Method of fabricating an optical-fiber-compatible sensor

ABSTRACT

A method for fabricating a sensor is provided, with the sensor including a reflective element and an optical fiber positioned relative to the reflective element such that light emitted from the optical fiber is reflected by the reflective element and propagates in an optical cavity between the optical fiber and the reflective element. The method includes positioning an element within the optical cavity. The element has a coefficient of thermal expansion and a thickness that compensate a refractive index change with temperature of a medium within the optical cavity.

CLAIM OF PRIORITY

This application is a divisional of U.S. patent application Ser. No.13/047,668, filed on Mar. 14, 2011 and incorporated in its entirety byreference herein, which claims the benefit of U.S. Provisional Appl. No.61/314,090, filed Mar. 15, 2010; U.S. Provisional Appl. No. 61/331,303,filed May 4, 2010; and U.S. Provisional Appl. No. 61/382,385, filed Sep.13, 2010, each of which is incorporated in its entirety by referenceherein.

BACKGROUND

1. Field of the Invention

This application relates generally to acoustic sensor systems, and moreparticularly to optical-fiber-compatible acoustic sensor systems.

2. Description of the Related Art

Various fiber optic sensor systems have been previously disclosed thatprovide acoustic pressure measurements based on the relativedisplacements of the two mirrors of a Fabry-Perot interferometriccavity. See, e.g., M. Yu et al., “Acoustic Measurements Using a FiberOptic Sensor System,” J. Intelligent Material Systems and Structures,vol. 14, pages 409-414 (July 2003); K. Totsu et al., “Ultra-MiniatureFiber-Optic Pressure Sensor Using White Light Interferometry,” J.Micromech. Microeng., vol. 15, pages 71-75 (2005); W. B. Spillman, Jr.et al., “Moving Fiber-Optic Hydrophone,” Optics Lett., vol. 5, no. 1,pages 30-31 (January 1980); K. Kardirvel et al., “Design andCharacterization of MEMS Optical Microphone for AeroacousticMeasurement,” 42nd AIAA Aerospace Sciences Meeting and Exhibit, 5-8 Jan.2004, Reno, Nev.; J. A. Bucaro et al., “Miniature, High Performance,Low-Cost Fiber Optic Microphone,” J. Acoust. Soc. Am., vol. 118, no. 3,part 1, pages 1406-1413 (September 2005); T. K. Gangopadhyay et al.,“Modeling and Analysis of an Extrinsic Fabry-Perot InterferometerCavity,” Appl. Optics, vol. 44, no. 16, pages 312-3196 (1 Jun. 2005);and P. J. Kuzmenko, “Experimental Performance of a Miniature Fabry-PerotFiber Optic Hydrophone,” Proceedings of 8th Optical Fiber SensorsConference, Monterey, Calif., Jan. 29-31, 1992, pages 354-357; O. Kilic,M. Digonnet, G. Kino, and O. Solgaard, “External fiber Fabry-Perotacoustic sensor based on photonic-crystal mirror,” in 18th InternationalOptical Fiber Sensors Conference, Cancun, Mexico (2006); O. Kilic, M.Digonnet, G. Kino, and O. Solgaard, “External fibre Fabry-Perot acousticsensor based on a photonic-crystal mirror,” Meas. Sci. Technol. 18,3049-3054 (2007); 0. Kilic, M. Digonnet, G. Kino, and O. Solgaard,“Photonic-crystal-diaphragm-based fiber-tip hydrophone optimized forocean acoustics,” in 19th International Optical Fiber SensorsConference, Perth, Australia (2008); 0. Kilic, M. Digonnet, G. Kino, andO. Solgaard, “Fiber-optical acoustic sensor based on a photonic-crystaldiaphragm,” in 15th International Conference on Solid-State Sensors,Actuators, and Microsystems, Denver, Colo. (2009).

Photonic-crystal slabs (PCSs) are photonic-crystal structures having aspatially periodically varying refractive index. A PCS exhibits guidedresonance optical modes that are strongly confined within the PCS, butare coupled to incident radiation through a phase matching mechanism dueto the periodically varying refractive index. These guided resonancemodes are typically manifest in transmission or reflection spectra assharp Fano lineshapes superimposed on a smoothly varying background.See, e.g., M. Kanskar et al., “Observation of leaky slab modes in anair-bridged semiconductor waveguide with a two-dimensional photoniclattice,” Appl. Phys. Lett., vol. 70, page 1438 (1997); V. N. Astratovet al., “Resonant coupling of near-infrared radiation to photonic bandstructure waveguides,” J. Lightwave Technol., vol. 17, page 2050 (1999);and S. Fan and J. D. Joannopoulos, “Analysis of guided resonances inphotonic crystal slabs,” Phys. Rev. B, vol. 65, page 235112 (2002). Suchguided resonance modes have been used previously as optical filters ormirrors in light emitting diodes and lasers

SUMMARY

In certain embodiments, an acoustic sensor is provided. The sensorcomprises a diaphragm comprising a reflective element. The sensor alsocomprises an optical fiber positioned relative to the reflective elementsuch that light emitted from the optical fiber is reflected by thereflective element. A first end of the optical fiber and the reflectiveelement forms an optical cavity therebetween. The sensor furthercomprises a structural element mechanically coupling the diaphragm andthe optical fiber. The structural element of certain embodimentscomprises a material having a coefficient of thermal expansionsubstantially similar to the coefficient of thermal expansion of theoptical fiber. For example, the structural element of certainembodiments comprises silica.

In certain embodiments, at least a portion of the light reflected by thereflective element can propagate into the optical fiber. The first endof the optical fiber can comprise a second reflective element. Thesecond reflective element and the reflective element can form aFabry-Perot cavity therebetween. In certain embodiments, the opticalfiber can comprise fused silica and the structural element can comprisefused silica. In some embodiments, the reflective element can comprise aphotonic-crystal structure. Additionally, the diaphragm of someembodiments can comprise silica. In various embodiments, the diaphragmof the acoustic sensor can have a thickness approximately equal to adistance between the first end of the optical fiber and the reflectiveelement.

In certain embodiments, the acoustic sensor can further comprise acompensating element comprising silica. The compensating element can bespaced from the diaphragm and positioned within the optical cavity. Thediaphragm of certain embodiments can have a lateral dimension and aratio of the lateral dimension to the optical fiber diameter can be in arange between 1.2 and 8. The diaphragm can have a movable portion havingan area and a ratio of the area to a cross-sectional area of the opticalfiber can be in a range between 1.4 and 64.

In certain embodiments, the diaphragm can comprise one or more fluidconduits. One or more fluid conduits can be separate from the reflectiveelement. In the acoustic sensor of certain embodiments, the opticalcavity can comprise a liquid. The acoustic sensor can further compriseat least one generally compressible and generally elastic element toincrease sensitivity. At least one generally compressible and generallyelastic element can be a gas bubble.

In certain embodiments, an acoustic sensor is provided. The sensorcomprises a reflective element. The sensor further comprises an opticalfiber positioned relative to the reflective element such that lightemitted from the optical fiber is reflected by the reflective element.The first end of the optical fiber and the reflective element form anoptical cavity therebetween. The optical cavity comprises a mediumhaving a refractive index change with temperature. In these embodiments,an element within the optical cavity has a coefficient of thermalexpansion and thickness that compensate the refractive index change withtemperature.

In various embodiments, the medium can be water. In these embodiments,the element within the optical cavity can comprise silica and can have athickness approximately equal to a distance between the first end of theoptical fiber and the reflective element. In some embodiments, theelement within the optical cavity can be a diaphragm mechanicallycoupled to the reflective element. The element within the optical cavitycan also be mechanically coupled to the optical fiber.

In certain embodiments, a method of fabricating an acoustic sensor isprovided. The method comprises providing a diaphragm. The diaphragmcomprises a reflective element. The method further comprises positioningan optical fiber relative to the reflective element such that lightemits from the optical fiber and is reflected from the reflectiveelement. Positioning the optical fiber relative to the reflectiveelement comprises forming an optical cavity therebetween. The methodfurther comprises mechanically coupling the diaphragm to the opticalfiber with a structural element. The structural element comprises amaterial having a coefficient of thermal expansion similar to thecoefficient of thermal expansion of the optical fiber. For example, thestructural element can comprise silica.

In certain embodiments, providing a diaphragm comprising a reflectiveelement can include providing a photonic-crystal structure as thereflective element. In these embodiments, providing a photonic-crystalstructure can comprise providing a photonic-crystal structure fabricatedby photolithography. In various embodiments, the method of fabricatingan acoustic sensor can further comprise silicate bonding the diaphragmto the structural element.

The method of fabricating an acoustic sensor can further compriseemploying an element comprising silica with the optical cavity. Incertain such embodiments, the method can further comprise selecting athickness for the element comprising silica approximately equal to adistance between the first end of the optical fiber and the diaphragm.The method of certain embodiments can comprise selecting a diaphragmdiameter to increase mechanical compliance. The method can furthercomprise selecting a diaphragm cross-sectional area to increasemechanical compliance.

In certain embodiments, the method can further comprise employing one ormore fluid conduits separate from the reflective element. In someembodiments, the method can further comprise employing at least onegenerally compressible and generally elastic element to increasesensitivity. At least one generally compressible and generally elasticelement can be a gas bubble.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A-1B schematically illustrate examples of acoustic sensorscompatible with certain embodiments described herein.

FIG. 2 is a plot of a portion of the response of an example acousticsensor as a function of wavelength for various temperatures.

FIG. 3 is an example plot of the calculated resonance wavelength changeas a function of temperature for a Fabry-Perot sensor comprising silicacompared to one comprising silicon.

FIGS. 4A-4B schematically illustrate examples of acoustic sensorscompatible with certain embodiments described herein.

FIG. 5 is a graph showing the variation of the temperature sensitivityof the optical path length with respect to different thicknesses of afused silica diaphragm in accordance with certain embodiments describedherein.

FIG. 6 shows the reflection spectrum calculated for an examplephotonic-crystal having a square pattern of holes with diameters of 800nm and a period of 900 nm, fabricated on a silicon diaphragm ofthickness 450 nm.

FIG. 7 shows the calculated change in reflectivity at 1550 nm as afunction of temperature for an example sensor in accordance with certainembodiments described herein.

FIG. 8 illustrates the contribution to the resonance wavelength changeas a function of temperature from various factors.

FIG. 9 schematically illustrates an example of an acoustic sensor withfluid conduits in accordance with certain embodiments described herein.

FIG. 10 shows the finesse of a fiber Fabry-Perot calculated for varyingreflectivities and cavity lengths in accordance with certain embodimentsdescribed herein.

FIGS. 11A-11B schematically illustrate example focusing elements inaccordance with certain embodiments described herein.

FIG. 12 schematically illustrates an example of an acoustic sensorcompatible with certain embodiments described herein.

FIG. 13A is an example response curve exhibiting cross-coupling for afirst sensor in parallel with a second sensor.

FIG. 13B is an example response curve exhibiting cross-coupling for asecond sensor in parallel with a first sensor.

FIG. 14A is an example response curve exhibiting a reduced orsubstantially eliminated cross-coupling for a first sensor in parallelwith a second sensor in accordance with certain embodiments describedherein.

FIG. 14B is an example response curve exhibiting a reduced orsubstantially eliminated cross-coupling for a second sensor in parallelwith a first sensor in accordance with certain embodiments describedherein.

FIGS. 15A-15E schematically illustrate an example photolithographyfabrication process in accordance with certain embodiments describedherein.

FIGS. 16A-16D schematically illustrate an example fabrication processfor producing a backside pattern in accordance with certain embodimentsdescribed herein.

FIGS. 17A-17C schematically illustrate example portions of threeindividual wafers and their patterns of holes to be used as buildingblocks of the silica structural element in accordance with certainembodiments described herein.

FIG. 18 schematically illustrates the wafers after they have been bondedtogether and attached to the photonic-crystal structure and the opticalfiber to form the sensor head in accordance with certain embodimentsdescribed herein.

FIG. 19 schematically illustrates the forces due to phenyl benzoate inaccordance with certain embodiments described herein.

FIGS. 20A-20B schematically illustrate structures used in a method thatreduces the arc current used to obtain a good bond between the twoelements in accordance with certain embodiments described herein.

FIG. 21 is a flowchart of an example method of fabricating an acousticsensor in accordance with certain embodiments described herein.

FIG. 22 schematically illustrates an example acoustic sensor fabricatedand assembled in accordance with certain embodiments described herein.

FIG. 23A shows a scanning electron micrograph of a top view of afabricated photonic-crystal mirror in accordance with certainembodiments described herein.

FIG. 23B shows a scanning electron micrograph of an angled view of afabricated photonic-crystal mirror in accordance with certainembodiments described herein.

FIG. 23C is a photograph of a fabricated sensor in accordance withcertain embodiments described herein.

FIG. 24 schematically illustrates an acoustic characterization setup totest example sensors.

FIG. 25 shows the measured coherence between a calibrated referencemicrophone and an example acoustic sensor.

FIG. 26 shows the measured frequency response of an example sensor.

FIG. 27 shows the measured noise (top curve), optoelectronic noise(middle curve), and the noise due to detection electronics (bottomcurve) for an example sensor.

FIG. 28 shows the measured minimum detectable pressure (MDP) of anexample sensor with the frequency response shown in FIG. 26.

FIG. 29 shows the measured thermal stability of the resonancewavelengths for a silicon sensor (top curve) and an example silicasensor (bottom curve).

FIG. 30 shows an example optical acoustic sensor system for oceanacoustics in accordance with certain embodiments described herein.

FIGS. 31A-31E schematically illustrate an example fabrication processfor producing a sensor system in accordance with certain embodimentsdescribed herein

FIG. 32 shows an optical profilometry measurement on an examplediaphragm in accordance with certain embodiments described herein.

FIG. 33 shows a photograph of an example packaged sensor system inaccordance with certain embodiments described herein.

FIG. 34 shows an example equivalent circuit formed by various lumpedelements describing sensor system acoustics and the interfacing withoptoelectronics in accordance with certain embodiments described herein.

FIG. 35A shows an example calculated response curve of a first sensor asa function of frequency calculated with the lumped-element model.

FIG. 35B shows the calculated noise spectrum (solid line) transferred toan example diaphragm showing contributions from radiation resistance(dashed line), hole resistance (dotted line), and channel resistance(dash-dotted line).

FIG. 35C shows the calculated noise spectrum (solid line) showingcontributions from the noise coupling from a second sensor (dashed line)and a third sensor (dotted line), and optoelectronic noise (dash-dottedline) in an example sensor system in accordance with certain embodimentsdescribed herein.

FIG. 36A shows the calculated minimum detectable pressure (MDP) as afunction of frequency on an example diaphragm in accordance with certainembodiments with the minimum ambient noise in the sea shown forreference.

FIG. 36B shows the minimum detectable pressure as a function offrequency when two parallel sensors are non-operational.

FIG. 37A shows the calculated linearity as a function of diaphragmdisplacement, showing the normalized linearities of the diaphragmdisplacement (solid line), Fabry-Perot response (dashed line), and powercoupled back into the optical fiber (dotted line).

FIG. 37B shows the total harmonic distortion (THD) as a function ofpressure amplitude for a first sensor (solid line), a second sensor(dashed line), and a third sensor (dotted line) in a sensor system inaccordance with certain embodiments described herein.

FIG. 38 show Reynolds number as a function of frequency for annularchannels for a first sensor (solid line), a second sensor (dashed line),and a third sensor (dotted line) in a sensor system in accordance withcertain embodiments described herein.

FIG. 39 schematically illustrates an example setup to characterizeexample sensor systems in accordance with certain embodiments describedherein.

FIG. 40 shows the coherence between a reference sensor system and anexample sensor system in accordance with certain embodiments describedherein.

FIG. 41A shows the measured frequency response for an example sensorsystem (solid line) and a theoretical fit (dashed line).

FIG. 41B shows the measured minimum detectable pressure (MDP) for anexample sensor system (solid line) and a theoretic fit (dashed line).

FIG. 42 shows the measure power spectrum of an example sensor system inaccordance with certain embodiments described herein.

DETAILED DESCRIPTION OF EXAMPLE EMBODIMENTS

Optical acoustic sensing has various important applications. Forexample, for structural health monitoring, acoustic sensors can monitorthe health of massive aerospace and wind-energy structures. Acousticsensors can also provide mobile detection, tracking, and reporting ofsubmarine contacts at long range in defense applications. As a furtherexample, the production from wells and zones within a well in oil andgas applications can be monitored and controlled. In yet anotherexample, acoustic sensing can measure the pressure of any bodily fluid,used in many medical applications, including life-support devices.

Certain embodiments described herein include structures, elements, orfeatures which advantageously address one or more issues arising frompreviously-disclosed acoustic sensors which utilize a mechanicaldiaphragm, a first reflective element, and a second reflective element(e.g., one or more photonic-crystal slabs) to produce a Fabry-Perotsensor with optical properties which are responsive to acoustic waves(e.g., acoustic waves incident on the sensor from the ambientenvironment or acoustic waves generated within the sensor). Certainembodiments described herein can be practiced by appropriatemodification of these previously-disclosed acoustic sensors. Examples ofsuch previously-disclosed acoustic sensors are described in U.S. Pat.No. 7,526,148, issued on Apr. 28, 2009; U.S. Pat. No. 7,630,589, issuedon Dec. 8, 2009; U.S. Pat. No. 7,809,219, issued on Oct. 5, 2010; andU.S. Pat. No. 7,881,565, issued on Feb. 1, 2011, each incorporated inits entirety by reference herein, and U.S. Pat. Appl. Publ. No.2011/0041616, which is also incorporated in its entirety by referenceherein. The structures, elements, or features described below can beused individually, or can be used with one another in combinations oftwo or more. Certain embodiments described herein can alternatively bepracticed by appropriate modification of other previously-disclosedacoustic sensor configurations (e.g., configurations not correspondingto those described in U.S. Pat. No. 7,526,148, U.S. Pat. No. 7,630,589;U.S. Pat. No. 7,809,219, U.S. Pat. No. 7,881,565, and U.S. Pat. Appl.Publ. No. 2011/0041616.

Reduced Sensitivity to Thermal Variation

FIGS. 1A-1B schematically illustrate example acoustic sensors 10 inaccordance with certain embodiments described herein. The acousticsensor 10 comprises a diaphragm 20 comprising a reflective element 22.In certain embodiments, the diaphragm 20 is deflectable by acousticwaves 400 and can comprise silicon, as is typically used in acousticsensors. In certain other embodiments, the diaphragm 20 canadvantageously comprise silica as will be discussed in more detailbelow. In yet other embodiments, the diaphragm 20 can comprise siliconnitride. Other materials are possible. The reflective element 22 ofcertain embodiments can be positioned (e.g., deposited) on the diaphragm20. In certain embodiments, the reflective element 22 can be bondeddirectly onto the diaphragm 20 (e.g., through thermal bonding). Incertain embodiments, the reflective element 22 is positioned (e.g.,deposited or bonded) on a surface of the diaphragm 20 facing away fromthe optical fiber 30, as shown in FIGS. 1A-1B. However, in otherembodiments, the reflective element 22 can be positioned (e.g.,deposited or bonded) on a surface of the diaphragm 20 facing towards theoptical fiber 30. In still other embodiments, the reflective element 22can be positioned (e.g., found) within the diaphragm 20. In variousembodiments, the diaphragm 20 comprises a reflective element 22comprising a photonic-crystal structure.

In certain embodiments, the reflective element 22 comprises a metallicmirror structure (e.g., one or more layers of gold, silver, aluminum,chromium, or combinations thereof). In certain embodiments, chromium,e.g., about 2 to about 5 nm thickness, can be used as an adhesion layerbeneath the reflective element 22. In these embodiments, the chromiumcan be relatively absorptive at certain wavelengths of interest. Thereflective element 22 can further comprise a thin (e.g., between about10 nanometers to about 100 nanometers thick) layer of silicon oxide toprotect the metal surface against oxidation and scratching.

In certain other embodiments, the reflective element 22 comprises adielectric mirror (e.g., multilayer structure comprising a plurality oftransparent dielectric layers with selected thicknesses and refractiveindices to provide a predetermined reflectivity). In certain suchembodiments, the dielectric mirror can have a thickness between 1 micronand 5 microns, and can have an area on the order of square inches (e.g.,a film stretched across a frame). Examples of dielectric materialscompatible with certain embodiments described herein include, but arenot limited to, silicon dioxide, magnesium fluoride, silicon monoxide,and tantalum oxide.

In certain other embodiments, the reflective element 22 comprises atleast a portion of a photonic crystal structure. The photonic crystalstructure of certain embodiments comprises one or more photonic crystalslabs. To make a photonic-crystal slab in accordance with certain suchembodiments, a dielectric layer, such as silicon or silicon nitride isdeposited on the outer surface of the diaphragm 20, and is subsequentlypatterned with holes going through the dielectric layer. An exampleprocess compatible with certain embodiments described herein is morefully discussed in U.S. Pat. No. 7,526,148, U.S. Pat. No. 7,630,589;U.S. Pat. No. 7,809,219, U.S. Pat. No. 7,881,565, and U.S. Pat. Appl.Publ. No. US2011/0041616, each of which is incorporated in its entiretyby reference herein.

The acoustic sensor 10 further comprises an optical fiber 30 positionedrelative to the reflective element 22 such that light emitted from theoptical fiber 30 is reflected by the reflective element 22. The opticalfiber 30 of certain embodiments is a single-mode fiber. Examplescompatible with certain embodiments described herein include, but arenot limited to, silica-based fiber, SMF-28® fiber available from CorningIncorporated of Corning, N.Y., cutoff shifted fiber, low-water-peakfiber, dispersion-shifted fiber, non-zero dispersion-shifted fiber, andnon-standard microstructured fiber (e.g., photonic crystal fiber).

As schematically illustrated by FIGS. 1A-1B, the optical fiber 30comprises a reflective element 32 (e.g., the first end 32 of the opticalfiber 30), and the reflective element 22 and the reflective element 32of the optical fiber 30 form an optical cavity 40 therebetween. Thereflective element 32 of the optical fiber 30 and the reflective element22 are spaced from one another in certain embodiments by a distancebetween 500 nanometers and 50 microns. In certain embodiments, sensors10 with a smaller optical cavity 40 can have a more advantageous thermalstability. In certain embodiments, the optical cavity 40 comprises a gas(e.g., air), while in certain other embodiments, the cavity 40 comprisesa liquid (e.g., water).

In certain embodiments, the optical fiber 30 transmits light from alight source to irradiate at least a portion of the reflective element22. Examples of light sources compatible with certain embodimentsdescribed herein include, but are not limited to, monochromatic sources(e.g., laser, laser diode), broadband sources (e.g., incandescent lamp,light-emitting diode), and tunable sources (e.g., tunable laser).

In certain embodiments, the reflective element 32 of the optical fiber30 comprises a metal layer at or on a first end of the optical fiber 30which is partially reflective and partially transmissive to lightemitted from the optical fiber 30. In certain embodiments, the metallayer comprises multiple sublayers of various materials, examples ofwhich include, but are not limited to, chromium, gold, silver, aluminum,and combinations thereof. In certain such embodiments, the metal layerfurther comprises a thin (e.g., between about 10 nanometers to about 100nanometers thick) layer of silicon oxide to protect the metal surfaceagainst oxidation and scratching. In certain embodiments, the metallayer has a thickness in a range between 1 nanometer and 50 nanometers.In certain other embodiments, the reflective element 32 of the opticalfiber 30 comprises a dielectric mirror at or on the first end of theoptical fiber 30 comprising a plurality of dielectric material layers.Examples of dielectric materials compatible with certain embodimentsdescribed herein include, but are not limited to, magnesium fluoride,zinc sulfide, silicon dioxide, titanium dioxide, and tantalum pentoxide.In certain embodiments, the reflective element 32 of the optical fiber30 comprises a photonic crystal structure at or on the first end of theoptical fiber 30.

In embodiments where the reflective element 32 of the optical fiber 30comprises a partially reflective end of the optical fiber 30, the end ofthe optical fiber 30 and the reflective element 22 of the diaphragm 20define a Fabry-Perot optical cavity 40 therebetween. After lightpropagates out of the optical fiber 30, at least a portion of the lightreflected by the reflective element 32 propagates back into the opticalfiber 30. As an incident acoustic wave 400 deflects the diaphragm 20, afrequency shift in the Fabry-Perot reflection spectrum can be induced.This shift can be detected as a change in the power reflected by theFabry-Perot optical cavity 40 at a fixed wavelength.

In certain embodiments, one or more factors, other than the incidentacoustic field that deflects the diaphragm 20 and affects the length ofthe optical cavity 40, can induce a frequency shift in the Fabry-Perotspectrum, and therefore an error in the measured acoustic pressure canoccur. For example, if the temperature of the Fabry-Perot optical cavity40 slowly increases, the material surrounding the Fabry-Perot opticalcavity 40 can expand. Hence, the spacing of the Fabry-Perot opticalcavity 40 can increase, and the reflection spectrum can slowly shift. Incertain embodiments, this frequency shift can be indistinguishable froma slow change in acoustic pressure. Furthermore, since at the probingwavelength the rate of change of the reflected power with wavelength (oroptical frequency) can change as the spectrum shifts, the responsivityof certain embodiments of the acoustic sensor 10 to a given displacementof the reflective element 22 can also change.

This process is illustrated in FIG. 2, which plots a portion of theresponse (reflected power/incident power) of an example acoustic sensor10 as a function of wavelength for various temperatures. As thetemperature is increased and the spectrum shifts (to the left in FIG.2), the operating (or bias) point at the laser wavelength (which isfixed), represented by a dot, shifts from a steep portion of the curvein the rightmost spectrum (for highest sensitivity) to a less steepportion. In certain embodiments, this can be undesirable because thescale factor of the acoustic sensor 10, which is proportional to theslope of the curve, decreases, e.g., the calibration of the sensorresponse to an acoustic field decreases. This can also be undesirablebecause the scale factor, and hence the response, can vary. Because itcan vary in an unpredictable manner, the ability of the acoustic sensor10 to perform calibrated measurements of acoustic power can becompromised. An important environmental parameter that results in suchan extraneous spectrum shift is temperature. As explained above, avariation in the temperature of the medium in which the acoustic sensor10 is located can shift the spectrum. In particular, if the thermalexpansion coefficients of the optical fiber 30 and the materialsurrounding the optical cavity 40 are different, the optical cavity 40can experience a length change with temperature variation.

Thus, certain embodiments described herein advantageously utilize astructural element mechanically coupling the diaphragm 20 with theoptical fiber 30 and comprising a material having a similar coefficientof thermal expansion as the optical fiber 30. For example, in certainembodiments as shown in FIGS. 1A-1B, the acoustic sensor 10 comprises astructural element 50 mechanically coupling the diaphragm 20 and theoptical fiber 30 and surrounding the optical cavity 40, wherein thestructural element 50 advantageously comprises a material having asimilar coefficient of thermal expansion as the optical fiber 30. Incertain embodiments, as shown in FIG. 1B, the structural element 50 caninclude a plurality of elements. Additionally, in certain embodiments aswill be described more fully below, the structural element 50 caninclude one or more holes, fluid conduits, or channels 55.

In certain embodiments, the optical fiber 30 is made of fused silica,which has a small thermal expansion coefficient (e.g., α=0.55×10⁻⁶/°C.), and the structural element 50 also comprises fused silica. By usinga structural element 50 made of the same low-thermal-expansion materialas is the optical fiber 30, the acoustic sensor 10 is renderedsubstantially insensitive to variations in ambient temperature. Incertain embodiments, as will be discussed below, the optical fiber 30can be inserted within a capillary tube. In various embodiments, thecapillary tube can advantageously comprise a material having a similarcoefficient of thermal expansion as the optical fiber 30. For example,the material can be silica.

As shown in FIG. 1B, the acoustic sensor 10 of certain embodimentsfurther comprises a housing 60 substantially surrounding the diaphragm20 comprising a reflective element 22, the structural element 50, theoptical cavity 40, and the optical fiber 30. In certain embodiments, thehousing 60 can comprise a plurality of elements, e.g., a protectivemembrane 61 and a backchamber housing 62. The protective membrane 61 cankeep the reflective element 22 and the optical cavity 40 isolated fromthe environment, e.g., to keep contaminants away and to preventcorrosion. The protective membrane 61 can be configured to allowacoustic waves 400 to propagate across the membrane 61 to deflect thediaphragm 20 (e.g. the membrane 61 can comprise a flexible, polymericmaterial).

In certain embodiments, the backchamber housing 62 can surround abackchamber or reservoir 65 that is in fluidic communication with theoptical cavity 40. It can be mechanically coupled to both the structuralelement 50 and the optical fiber 30, as shown in FIG. 1B. In someembodiments, the backchamber housing 62 comprises brass or aluminum. Inother embodiments, the backchamber housing 61 advantageously comprises amaterial with a similar coefficient of thermal expansion as the opticalfiber 30 and/or structural element 50 for similar reasons discussedabove. Thus, the backchamber housing 62 can comprise silica.

FIG. 3 is an example plot of the resonance wavelength change as afunction of temperature for a Fabry-Perot sensor 10 with a sensor headcomprising silica in accordance with certain embodiments describedherein compared to one comprising silicon, using a probing wavelength of1550 nm. As shown in FIG. 3, the all-silica sensor 10 (e.g., silicafiber 30, silica capillary tube, and silica structural element 50) ofcertain embodiments offers a substantial enhancement in thermalstability as compared to the sensor comprising the silica fiber and thesilicon sensor head.

In certain embodiments, the effects of thermal expansion on thesensitivity of the acoustic sensor 10 are at least a factor of 10smaller than other effects on the sensitivity of the acoustic sensor 10.Simulations show that with suitable design, the sensitivity of certainembodiments of the acoustic sensor 10 does not change by more than 10%for a temperature variation of greater than 100° C. Assuming that theFabry-Perot cavity 40 is filled with air, for a Fabry-Perot cavity 40with a 10-μm mirror spacing and a finesse of 30, the temperature changethat changes the sensitivity of the sensor 10 by 10% is 300° C. Thefinesse F of a Fabry-Perot cavity is defined as F=2πN, where N is thenumber of round trips when the loss factor becomes 1/e. In other words,the energy inside the cavity drops to 1/e of its initial value after Nround trips. The temperature change is approximately inverselyproportional to the finesse, so that, e.g., a sensor 10 having anair-filled Fabry-Perot cavity 40 with a mirror spacing of 10 μm and afinesse of 300 can tolerate a maximum temperature change of around 30°C. for a sensitivity variation of no more than 10%.

For a Fabry-Perot cavity 40 containing water, thermal variations of therefractive index of water can have further detrimental effects on theperformance of certain embodiments of the optical-fiber-compatibleacoustic sensor 10. In certain embodiments in which the sensor 10 ofFIG. 1B is employed in water as a hydrophone, the Fabry-Perot cavity 40is filled with water. This water can be either the ambient water inwhich the sensor 10 or hydrophone head is immersed, or a separatereservoir of water isolated from the ambient water by an enclosure, suchas the protective membrane 61. The refractive index of water varies withtemperature, more so than does the refractive index of air, and itseffect on the thermal sensitivity of the sensor 10 is about one order ofmagnitude larger than the effect of the thermal expansion of silica (thedn/dT coefficient of water is −11.8×10⁻⁶/° C. for optical wavelengthsaround 1550 nm).

In a sensor 10 as schematically illustrated by FIGS. 1A-1B, the maximumtolerable temperature change for a cavity filled with water is generallysmaller by a factor of 15 than the maximum tolerable temperature changefor a cavity filled with air. For example, for a 10-μm water-filledFabry-Perot cavity 40 with a finesse of 30, the temperature change whichchanges the sensitivity of the sensor 10 by 10% is only 20° C. Thistemperature change is approximately inversely proportional to thefinesse, so that, e.g., a sensor 10 having a cavity 40 with a finesse of300 can tolerate a maximum temperature change of only 2° C. for asensitivity variation of no more than 10%.

Certain embodiments described herein advantageously compensate for therefractive index change of water with temperature. FIG. 4A schematicallyillustrates an example of an acoustic sensor 10 compatible with certainembodiments described herein. The acoustic sensor 10 comprises areflective element 22. The acoustic sensor 10 further comprises anoptical fiber 30 positioned relative to the reflective element 22 suchthat light emitted from the optical fiber 30 is reflected by thereflective element 22. The reflective element 32 of the optical fiber 30and the reflective element 22 define an optical cavity 40 therebetween.The optical cavity 40 comprises a medium having a refractive indexchange with temperature. The acoustic sensor 20 further comprises acompensating element 25 positioned within the optical cavity 40 andhaving a coefficient of thermal expansion and thickness. In certainembodiments, the coefficient of thermal expansion and the thickness areselected such that the compensating element 25 compensates therefractive index change with temperature of the medium. In certain suchembodiments, this compensation is sufficient for the optical sensor tohave reduced thermal variation in performance as compared to an opticalsensor without the compensating element.

The compensating element 25 can comprise one or more pieces of materialthat are selected to provide a coefficient of thermal expansion andtotal thickness so that the sensor 10 has a reduced sensitivity totemperature variations. As shown in FIG. 4A, the compensating element 25within the optical cavity 40 comprises of material spaced away from thediaphragm 20. Such a material can be part of the first end 32 of theoptical fiber 30. For example, the material can be attached to thereflective end of the optical fiber 30 before the optical fiber 30 isinserted into the sensor head. In certain embodiments in which thereflective element 32 of the optical fiber 30 is spaced along theoptical fiber 30 and away from the end of the optical fiber 30, thecompensating element 25 can comprise the portion of the optical fiber 30between the reflective element 32 and the end of the optical fiber 30.Alternatively, at least a portion of the compensating element 25 can beformed by micro-fabrication such that it is positionable partway betweenthe reflective element 32 of the optical fiber 30 and the reflectiveelement 22. For example, at least a portion of the compensating element25 can be on the diaphragm 20 facing the optical fiber 30 (or can bemechanically coupled to another portion of the optical sensor 10 (e.g.,the structural element 50).

In certain embodiments, as shown in FIG. 4B, at least a portion of thediaphragm 20 can serve as the compensating element 25 within the opticalcavity 40. In accordance with certain such embodiments in which thediaphragm 20 comprises silica, the compensating element 25 within theoptical cavity 40 has a thickness (labeled S in FIG. 4B) substantiallyequal to the spacing between the fiber end 32 and the diaphragm 20(labeled W in FIG. 4B). The reflective element 22 can be a materialcoated or fabricated on the diaphragm 20. The spacing volume is filledwith water, and light is reflected from the reflective element 22 on theside of the diaphragm 20 facing away from the optical fiber 30. Asdiscussed above, the reflective element 22 can comprise layers ofmetals, dielectrics, or photonic-crystal structure formed, deposited, orbonded on the diaphragm 20.

For a given temperature change, the refractive index of fused silicachanges by approximately the same magnitude as for water, but in theopposite direction (the dn/dT coefficient of fused silica is about+12.8×10⁻⁶/° C. for optical wavelengths around 1550 nm while dn/dT forwater is about −12.8×10⁻⁶/° C. for these optical wavelengths).Therefore, in certain such embodiments, when light propagates byapproximately equal distances through water and silica, the temperatureeffect on the refractive index of water is effectively cancelled out bythe temperature effect on the refractive index of silica. FIG. 5 is agraph showing the variation of the temperature sensitivity of theoptical path length (physical length multiplied by the refractive index)with respect to different thicknesses of the fused silica diaphragm 20.FIG. 5 corresponds to the spacing between the reflective tip of thefiber 32 and the diaphragm 20 (labelled “W” in FIG. 4B) being heldconstant at 10 μm, and the diaphragm thickness (labelled “S” in FIG. 4B)being varied from 6 μm to 10 μm, with the total optical thickness(T=S+W) being varied from 16 μm to 20 μm. The absolute value of thetemperature sensitivity of the optical path length the light travels inthe cavity versus diaphragm thickness (plotted as the solid curve) issignificantly below the absolute value of the temperature sensitivityfor a non-silica diaphragm (shown in FIG. 5 as the dash-dot line) and isbelow a maximum practical temperature sensitivity (shown in FIG. 5 asthe dotted line) for the entire range of diaphragm thicknesses betweenabout 6.15 μm and 10 μm. A minimum temperature sensitivity is observedfor a diaphragm thickness of about 8.15 μm, corresponding to a sensor 10in which the refractive index variations and material expansionscompensate each other, such that the sensor 10 is rendered substantiallyinsensitive to temperature variations. In certain embodiments, for apractical range of diaphragm thicknesses, a sensor 10 or hydrophonehaving a water-filled cavity and employing a silica diaphragm 20 is evenless sensitive to temperature than is the sensor 10 upon having anair-filled cavity (shown in FIG. 5 as the dashed line). The relationshipbetween the temperature sensitivity dn/dT of the optical path lengthwith respect to different thicknesses for the compensating element canbe determined for other materials for the compensating element and forother media for the optical cavity.

In certain embodiments, the diaphragm thickness is selected to render asensor with a water-filled cavity substantially insensitive to thermaleffects. For example, in certain embodiments in which the sensorcomprises a 10-μm water-filled cavity between the diaphragm 20 and theoptical fiber 30, the diaphragm thickness is in a range between about 5μm and about 12 μm, between about 7 μm and about 10 μm, or between about8 μm and about 9 μm. In certain embodiments, the ratio of the thicknessof the diaphragm 20 to the cavity size between the diaphragm 20 and theoptical fiber 30 is in a range between about 0.5 and about 1.2, betweenabout 0.7 and about 1, or between about 0.8 and about 0.9. The value ofthe diaphragm thickness of 8.15 μm denoted in FIG. 5 for the 10-μmwater-filled cavity is based on an assumption that the light is directlyreflected from the reflective element 22 on the outer surface of thediaphragm 20. This assumption is accurate for certain embodiments whenmetal layers are used as the reflective element 22. When dielectricmirrors or photonic crystals (which can range in thickness approximatelyfrom 0.5 μm to 5 μm) are used, however, light travels beyond the outersurface of the diaphragm 20 into the reflective element 22 before it isreflected. Therefore, to compensate for thermal expansion and refractiveindex changes with temperature of the reflective element 22, thediaphragm thickness can be adjusted to obtain the optimum temperatureinsensitivity for a given reflective element 22.

Because the mechanical compliance of a thick diaphragm 20 (e.g., athickness of 8.15 μm) is low, it can be difficult to deflect such adiaphragm 20 in certain embodiments. In certain embodiments, this issuecan be resolved by increasing the diameter of the diaphragm 20 toincrease the mechanical compliance, as described more fully below.

Another issue regarding the example configuration schematicallyillustrated in FIG. 4B could be the reflection from the surface 21 ofthe silica diaphragm 20 facing towards the optical fiber 30. However,due to the small difference between the refractive indices of silica andwater (n_(silica)=1.444 vs. n_(water)=1.316 at 1550 nm wavelength), thereflection (R) from a silica-water interface, hence from the diaphragmsurface 21, can be negligible (R<0.3%). In certain embodiments, thisreflection can also be eliminated or reduced sufficiently by depositingan anti-reflection coating on the surface 21 of the diaphragm 20.

In embodiments where the reflective element 22 comprises aphotonic-crystal mirror, the thermal response of the photonic-crystalmirror is another factor that affects the thermal stability of thesensor 10. As the temperature changes, the refractive index of thematerials of the photonic-crystal mirror change, and so do its physicaldimensions, (e.g., the thicknesses of the materials, and the periodicityand the diameter of the periodic structures, such as holes). Since allof these parameters affect the reflection spectrum of thephotonic-crystal mirror, as these parameters change, the spectrum alsochanges. As a result of the change in the reflectivity of thephotonic-crystal mirror, the finesse of the Fabry-Perot optical cavity40 changes, and so does the slope of its reflection spectrum, inparticular at the optimum bias point shown in FIG. 2, and the scalefactor of the sensor 10.

Finite-difference-time-domain (FDTD) simulations of the effect oftemperature on the reflection spectrum of the photonic-crystal mirrorshow that this contribution is small for certain application. Forexample, FIG. 6 shows the reflection spectrum calculated for an examplephotonic-crystal structure having a square pattern of holes withdiameters of 800 nm and a period of 900 nm, fabricated on a silicondiaphragm 20 of thickness 450 nm. These parameters were selected toobtain a high reflection at 1550 nm, a convenient target wavelength forthis type of sensor. This photonic-crystal design provides ˜99%reflectivity at 1550 nm and a bandwidth of 48 nm for 99% reflectivity.

Using the same FDTD method, the spectrum of the same photonic-crystalstructure can be simulated at different temperatures, taking intoaccount the changes in refractive index, in hole radius, in period, andin thickness of the diaphragm. FIG. 7 shows the calculated change inreflectivity at 1550 nm as a function temperature for a sensor 10 inaccordance with certain embodiments described herein. For apredetermined range of temperatures, e.g., from about 20° C. up to amaximum simulated temperature of about 80° C., the reflectivity remainswithin 0.02% of its value at 20° C. The bandwidth of thephotonic-crystal structure for 99% reflectivity, not shown in FIG. 7,remains within 2.1% over this temperature range. In certain embodiments,the reflectivity remains within 0.03%, 0.04%, 0.05%, 0.08%, or 0.10% ofits value at 20° C. over a range of temperature of about 20° C. to about80° C.

The result of this small variation in the photonic-crystal reflectivityis that the resonance wavelength of the sensor remains within 0.02 nmover a 400° C. temperature range assuming a 90% reflectivity for thereflecting element 32 at the end 32 of the optical fiber 30, whichtranslates into a nominal finesse for the Fabry-Perot optical cavity of96.

Another contribution to the thermal instability of the Fabry-Perot-basedacoustic sensor 10 is thermally induced variations in the refractiveindex of the optical cavity 40, e.g., the intra-cavity medium. When thismedium is air, as in the case of a microphone for example, thiscontribution can be negligible. However, when it is water, as may be thecase in a hydrophone, a change in this refractive index can induce anadditional shift in the resonance of magnitude:

$\begin{matrix}{\left( \frac{\Delta \; \lambda}{\lambda} \right)_{RIM} = \frac{\Delta \; n}{n}} & (1)\end{matrix}$

In the case of water, the shift in resonance wavelength due to thiseffect stays within ±1 nm, thus provides enough stability over ±100° C.before the maximum responsivity drops by more than 10% for a Fabry-Perotcavity of length 10 μm. This shift can be acceptable for manyapplications.

FIG. 8 illustrates the contribution from each individual factordescribed above: thermal expansion of silica (TE), thermally inducedvariation of the intra-cavity medium refractive index (RIM), andthermally induced variation in the spectral response of thephotonic-crystal mirror (PC). FIG. 8 also shows the resonance wavelengthchange with temperature resulting from the sum of these three effects.Because the intra-cavity medium is taken to be water in this analysis,and because water has a negative thermo-optic coefficient, thecontribution of the intra-cavity medium refractive index can benegative, e.g., its sign is opposite that of the other twocontributions, hence it partially cancels them. A different choice ofmaterials and/or design parameters could tailor the amount ofcancellation and total contribution.

In certain embodiments, the material for the medium of the opticalcavity 40 can be advantageously selected for improved thermal stability.In addition to the length of the optical cavity 40, the thermalmodulation of the refractive index of the medium of the optical cavity40 also can contribute to the thermal stability of the sensor 10. Forexample,

$\begin{matrix}{L = \left. {q\; \frac{\lambda}{2\; n}}\Rightarrow{\lambda \propto {{nL}.}} \right.} & (2)\end{matrix}$

For no resonance shift:

$\begin{matrix}{{{- \frac{1}{n}}\frac{\partial n}{\partial T}} = {\alpha_{{SiO}_{2}} = {0.55 \times {10^{- 6}/{^\circ}}\mspace{14mu} {C.}}}} & (4)\end{matrix}$

where L is the length of the optical cavity 40, n is the refractiveindex of the cavity medium, and α_(siO2) is the thermal expansioncoefficient of silica. In certain embodiments, this effect can beexploited for thermal stability. For example, in various embodiments,the effect of the thermal expansion of the silica structural element 40and the refractive index modulation of the medium of the optical cavity40 cancel each other if the right material is selected for the cavitymedium. For example,

$\begin{matrix}{{\frac{\partial({nL})}{\partial T} = {\left. 0\Rightarrow{{n\frac{\partial L}{\partial T}} + {L\frac{\partial n}{\partial T}}} \right. = {{{{nL}\; \alpha_{{SiO}_{2}}} + {L\frac{\partial n}{\partial T}}} = 0}}}{{n\; \alpha_{{SiO}_{2}}} = {\left. {- \frac{\partial n}{\partial T}}\Rightarrow\alpha_{{SiO}_{2}} \right. = {{- \frac{1}{n}}\frac{\partial n}{\partial T}}}}} & (3)\end{matrix}$

Thus, in certain embodiments, the medium for the optical cavity 40 canbe selected for improved thermal stability.

Increased Lateral Dimension or Area of the Diaphragm

As mentioned above, a thicker diaphragm 20 is generally mechanicallyless compliant than is a thinner diaphragm 20. In addition, one of thestrongest damping effects that can limit the sensitivity of the sensor10 is squeeze-film damping, which is due to the water forced out of thecavity 40 by the moving diaphragm 20, as is described more fully in U.S.Pat. No. 7,526,148, U.S. Pat. No. 7,630,589; U.S. Pat. No. 7,809,219,U.S. Pat. No. 7,881,565, and U.S. Pat. Appl. Publ. No. 2011/0041616,each of which is incorporated in its entirety by reference herein.

Certain embodiments described herein restore the compliance of thediaphragm 20 by increasing the diaphragm diameter (e.g., byapproximately a factor of 5) or the diaphragm area (e.g., byapproximately a factor of 25). Such a significant increase in thediaphragm diameter or area also reduces the squeeze-film dampingsignificantly (e.g., by approximately a factor of 25), since therelative area of the end face of the optical fiber 30 to the area of thediaphragm 20 is reduced. In certain embodiments, the ratio of thediaphragm diameter to the end diameter of the optical fiber 30 is in arange between 1.2 and 8, in a range between 1.5 and 6, or in a rangebetween 2 and 5. In certain embodiments, the ratio of the diaphragm areato the area of the end face of the optical fiber 30 is in a rangebetween 1.4 and 64, in a range between 2.35 and 36, or in a rangebetween 4 and 25. For example, for a diaphragm diameter of about 300 μmand a fiber end diameter of about 125 μm, the diameter ratio is about2.4 and the area ratio is about 5.76. However, by increasing thediaphragm diameter to about 600 μm, the diameter ratio is about 4.8 andthe area ratio is about 23, resulting in a reduction of the squeeze-filmdamping by about a factor of 23. In certain embodiments, the diaphragmdiameter or area is limited by the desired resonance frequency of thediaphragm 20. For example, in certain embodiments in which higherfrequencies are to be detected, the diaphragm diameter is less than 1mm. The use of the diaphragm diameter in describing this feature is notintended to indicate that the diaphragm shape is limited to solelygenerally circular diaphragms. Other diaphragms having other shapes(e.g., oval, square, octagon, or other polygonal or irregular shapes)may also be used in accordance with certain embodiments describedherein. In these embodiments, the diaphragm 20 has a lateral dimensionand the compliance of the diaphragm 20 can be restored by increasing thediaphragm lateral dimension as described above. In these embodiments,the compliance of the diaphragm 20 can be restored by increasing thecross sectional area of the diaphragm 20.

Pressure-Equalization Channels

As discussed above, the reflective element 22 (e.g., a reflectivesurface on the outside of the diaphragm 20) of certain embodiments canbe a dielectric- or metal-based mirror, or a photonic-crystal reflector.As described in U.S. Pat. No. 7,526,148, U.S. Pat. No. 7,630,589; U.S.Pat. No. 7,809,219, U.S. Pat. No. 7,881,565, and U.S. Pat. Appl. Publ.No. 2011/0041616, each of which is incorporated in its entirety byreference herein, a photonic-crystal mirror reflector can also serve asthe mechanical diaphragm 20 comprising a reflective element 22. Besidesserving to provide the refractive index and periodicity of thephotonic-crystal structure, the holes extending through the diaphragm 20in certain such embodiments can serve as pressure-equalization channelsas well, to allow the hydrostatic pressures between the outside andinside of the sensor 10 to equalize. However, using the same holes totailor the optical properties of the photonic-crystal reflector, themechanical compliance of the diaphragm 20, and the acoustic response ofthe sensor 10 at low frequencies can create challenges in designing theoptimum sensor 10 for a given application.

In certain embodiments, this issue can be alleviated wholly or in partas follows. A set of one or more fluid conduits (e.g., holes) is formed(e.g., by etching or drilling) in the sensor 10 to allow fluid flow fromone side of the diaphragm 20 to the other for pressure equalizationacross the diaphragm 20. In certain embodiments, as shown in FIG. 9, oneor more of the fluid conduits 55 can be through the diaphragm 20. Forexample, the one or more fluid conduits 55 can be through a diaphragm 20sufficiently thick to reduce the sensitivity to thermal effects asdescribed above, or through a thicker diaphragm 20 that is mechanicallyless compliant as described above.

In certain embodiments, one or more of the fluid conduits 55 areseparate from the photonic-crystal structures of the diaphragm 20 (e.g.,holes in a thick diaphragm 20 as described above) which affect theoptical properties of the reflector or reflective element 22. Forexample, in certain such embodiments, one or more of the fluid conduits55 are located in a portion of the diaphragm 20 which does notcontribute to the optical properties of the Fabry-Perot cavity 40, e.g.,separate from the reflective element 22. In certain other embodiments,as shown in FIG. 1B, one or more of the fluid conduits 55 are separatefrom the diaphragm 20 (e.g., conduits through or along a portion of thestructural element 50). In some embodiments, the sensor 10 can includeone or more fluid conduits 55 in both the diaphragm 20 and thestructural element 50. In certain embodiments, the total cross-sectionalarea of the set of one or more fluid conduits is in a range betweenabout 1 μm² and about 50 μm². In certain embodiments, the totalcross-sectional area of the one or more fluid conduits is sufficientlysmall such that, at the desired operational acoustic frequency range,the fluid (e.g., water) preferably moves through the one or more fluidconduits rather than through the photonic-crystal structures (e.g.,holes).

Certain embodiments described herein allow the optical and acousticdesign constraints to be separately satisfied, thereby allowing bettersensor optimization. For example, by having one or more fluid conduits55 which are separate from the photonic-crystal holes which provide theoptical properties of the photonic-crystal reflective element 22, otherphotonic-crystal reflector structures can be used which do not provide afluid conduit for fluid flow across the diaphragm 20 (e.g.,photonic-crystal structures with protrusions rather than holes, orphotonic-crystal structures with holes that do not go through the fullthickness of the diaphragm 20). This method of separating the optical,mechanical, and acoustical design is not specific to a thick diaphragm20, and can also be employed for thinner diaphragms 20, whenever it isdesired to decouple the mechanical and acoustical functions from theoptical function of the photonic-crystals structures (e.g. holes).

Reduced Diffraction Losses

In certain embodiments, the thicker diaphragm 20 described above (e.g.,the diaphragm 20 sufficiently thick to reduce the sensitivity to thermaleffects, or the thicker diaphragm 20 that is mechanically lesscompliant) can result in an increase of the optical path length betweenthe first end 32 of the optical fiber 30 and the reflective element 22,which can cause additional diffraction loss. Unless counteracted in someway, this additional diffraction loss can reduce the reflectivity, andhence the sensitivity of the sensor 10.

FIG. 10 shows the finesse of a fiber Fabry-Perot cavity 40 (e.g., asdepicted in FIG. 4B) as a function of reflectivity and for variouscavity lengths in accordance with certain embodiments described herein.The finesse of the fiber Fabry-Perot cavity 40, which can be termed the“effective finesse,” includes the effect of diffractive loss of energywhich is not coupled back into the optical fiber 30. The curves of FIG.10 were calculated for a Fabry-Perot cavity 40 formed by an SMF-28single-mode fiber 30 and a reflective element 22, and by varying boththe cavity length 40 and the reflectivities of the reflective element22. See, e.g., Kilic et al., “Asymmetrical Spectral Response in FiberFabry-Perot Interferometers,” J. Lightwave Technology, vol. 27, no. 24,pages 5648-5656 (2009). The solid line of FIG. 10 corresponds to thecalculated finesse as a function of reflectivity for a standardFabry-Perot cavity between two planar and infinite reflective surfaces.For larger cavity 40 lengths, as would be the case for a thick diaphragm20, the finesse is dominated by diffraction loss, and is therefore notaffected much by the reflectivities of the reflective element 22 (see,e.g., the lines corresponding to cavity lengths of 8%, and 16λ). Sincethe sensor sensitivity is proportional to finesse, a high finesse isdesirable to improve the sensitivity of the sensor 10 (e.g., by reducingthe diffraction loss).

In certain embodiments, the sensor 10 comprises a focusing element 70(e.g., a lens or curved mirror) as part of the optical path of theFabry-Perot cavity 40 in order to reduce diffraction loss. FIGS. 11A-11Bschematically illustrate two example focusing elements 70 in accordancewith certain embodiments described herein. FIG. 11A schematicallyillustrates a diaphragm 20 comprising a lens structure 70 (e.g., acurved surface fabricated as at least a part of the surface of thediaphragm 20 facing towards the optical fiber 30). FIG. 11Bschematically illustrates a diaphragm 20 comprising a curved reflectivesurface or layer 70 (e.g., a curved mirror fabricated as at least a partof the surface of the diaphragm 20 facing away from the optical fiber30). In certain embodiments, the curvatures of either the lens structureor the reflective surface of layer 70 can be chosen so that themode-field diameter of the light beam reflected back to the fiber's endface is matched to the mode-field diameter of the fiber mode, such thatthe diffraction loss can be substantially reduced or eliminated. Forexample, in certain embodiments, the radius of curvature of either thelens structure or the reflective surface of layer 70 is in a rangebetween about 0.1 mm and about 0.6 mm.

As schematically illustrated by FIGS. 11A-11B, the focusing element 70(e.g., the lens and/or curved mirror) of certain embodiments is a partof the diaphragm 20. In certain other embodiments, the focusing element70 is separate from the diaphragm 20 but is still part of the opticalpath of the Fabry-Perot cavity 40. For example, the focusing element 70can comprise a separate slab or structure spaced away from the diaphragm20 (e.g., a lens structure between the diaphragm 20 and the opticalfiber 30 or a structure positioned on the optical fiber 30). Otherconfigurations are also compatible with certain embodiments describedherein.

Improved Dynamic Range

FIG. 12 schematically illustrates an example of an acoustic sensorsystem 100 having a plurality of sensors compatible with certainembodiments described herein. Scanning electron micrographs of anexample backside wafer, diaphragm 20, and frontside wafer are shownbeneath the schematic. In this example, the structural element 50(comprising the backside wafer and the frontside wafer) is fabricatedwith silicon, and the reflective element 22 of the diaphragm 20comprises photonic-crystal mirrors positioned to form optical cavitieswith the two single-mode optical fibers 30.

In ocean acoustics, because water is practically incompressible, thediaphragm 20 may not move against a small close Fabry-Perot cavityfilled with water. Thus, channels 90, e.g., diaphragm-sized channels,can be fabricated around the fibers to allow water to flow out of theoptical cavity 40 and to allow the diaphragm 20 to move. In certainembodiments, the diaphragm-sized channels 90 are between about 0.1 mmand about 0.4 mm in diameter, between about 0.15 mm and about 0.35 mm indiameter, or between about 0.2 mm and about 0.3 mm in diameter. Incertain embodiments, the diaphragm-size channels 90 define the diametersof the diaphragms 20 and provide a connection around the optical fibers30 to expanded channels 92. The expanded channels 92 can further lead toa backchamber channel 95. In certain embodiments, the expanded channels92 are larger than the diaphragm-sized channels 90 to reduce flowresistance within the expanded channels 92. The backchamber channel 95can be a large hole at the center of the structural element 50. Incertain embodiments, the backchamber channel 95 is between about 1 mmand 2 mm in diameter, e.g., about 1.5 mm in diameter.

In certain embodiments, as shown in FIG. 12, two or more sensors 101,102 that are responsive to different acoustic signal levels can be usedin parallel with one another to improve the dynamic range of the sensorsystem 100. In certain such embodiments, the plurality of parallelsensors 101, 102 are placed close to each other, so that they areexposed to approximately the same acoustic signal. In certainembodiments utilizing two sensors (e.g., a first sensor 101 and a secondsensor 102), the first sensor 101 can be used to measure weak acousticsignals, and the second sensor 102 can be used to measure strongersignals. In this way, the total dynamic range of sensor system 100 withthe two combined sensors 101, 102 is larger than the dynamic range ofeither sensor 101 or 102 alone.

In certain embodiments, at least one sensor of the plurality of sensors(e.g., the second sensor 102 of the first and second sensors 101, 102)can measure stronger signals, but has a reduced sensitivity, as comparedto the other sensors (e.g., the first sensor 101) of the plurality ofsensors. In certain such embodiments, the sensitivity of at least onesensor is reduced by various methods, techniques, or modifications. Forexample, the finesse of the Fabry-Perot cavity 40 of the at least onesensor (e.g., the second sensor 102) can be reduced by using areflective element 22 having a lower reflectivity, by using a longerFabry-Perot cavity 40, or both. Such modifications of the Fabry-Perotcavity 40 cause a higher diffraction loss and thereby reduce the finesseof the Fabry-Perot cavity 40.

In certain other embodiments, the mechanical compliance of the diaphragm20 in the at least one sensor (e.g., the second sensor 102) can bereduced as compared to the other sensors (e.g., the first sensor 101).For example, a thicker diaphragm 20, and/or a diaphragm 20 with asmaller diameter, and/or a diaphragm 20 made of a less compliantmaterial can be used to reduce the mechanical compliance of thediaphragm 20.

In certain embodiments, at least one sensor can utilize an opticaldetection scheme different than that of a Fabry-Perot cavity 40. Forexample, at least one sensor can comprise a bare fiber 30 (e.g., a fiber30 without any reflective element 32 on its end), such that there is nosignificant reflection from its end face (since silica-water interfacereflection is less than 0.3%). The motion of the diaphragm 20 in certainsuch embodiments only affects the amount of light coupled back into theoptical fiber 30, since the coupling is dependent on the spacing betweenthe diaphragm 20 and the fiber end. The coupled signal, consequently,can be used in the same way the Fabry-Perot signal is used to measurethe acoustic signal.

Reduced Cross-Coupling Between Sensors

Due to the low compressibility of water, movement of the diaphragm 20 inresponse to an acoustic signal results in a flow of water in and out ofthe optical cavity 40. In certain embodiments, a reservoir, referred toas the backchamber 65, is provided inside the sensor 10. The backchamber65 comprises a volume of water (e.g., a few cubic millimeters in size)that is in fluidic communication with the optical cavity 40. When two ormore sensors 101, 102 are employed in parallel to increase the dynamicrange, as discussed above, the large size of the backchamber 65 may makeit impractical in some embodiments to employ separate backchambers 65for each sensor 101, 102. Therefore, in certain embodiments utilizingparallel sensors 101, 102, a single backchamber 65 can be shared bymultiple, or even all, sensors 101, 102. However, such a configurationin certain embodiments can allow cross-coupling of the signal and noisebetween the sensors 101, 102 sharing a backchamber 65.

FIGS. 13A and 13B are plots of the example responses of a first sensor101 (e.g., FIG. 13A) and a second sensor 102 (e.g., FIG. 13B) of a pairof two sensors 101, 102 in parallel with one another, sharing the samebackchamber 65. For this particular example of FIGS. 13A and 13B, (i)the first sensor 101 has a 0.5-μm-thick diaphragm 20 with a diameter of200 μm, and the resonance of the first sensor 101 is at 18 kHz, (ii) thesecond sensor 102 has a diaphragm 20 with the same thickness as thefirst sensor 101, but with a 180-μm-diameter and a resonance at 21 kHz,and (iii) the backchamber 65 is a cylindrical volume with a radius of 3mm and a length of 5 mm, and it has a Helmholtz resonance of 82 kHz.

As is evident from FIGS. 13A and 13B, the two sensors 101, 102 couple toeach other, and introduce additional resonant features. The arrow ofFIG. 13A points to a resonance feature in the response of the firstsensor 101 due to coupling from the signal of the second sensor 102, andthe arrow of FIG. 13B points to a resonance feature in the response ofthe second sensor 102 due to coupling from the signal of the firstsensor 101. This cross-coupling can be detrimental, in certainembodiments, to the sensor performance, since it complicates theresponse and couples noise between sensors 101, 102, so that the noisefloor for each sensor 101, 102 is increased.

Certain embodiments described herein advantageously eliminatecross-coupling between the two or more parallel sensors 101, 102. Incertain such embodiments, the Helmholtz resonance of the backchamber 65and the sensor resonances are tailored so that they are substantiallyequal in frequency with one another. In certain such embodiments, at theHelmholtz resonance, the impedance of the backchamber 65 is zero suchthat the two parallel sensors 101, 102 are acoustically grounded, henceuncoupled. Certain such embodiments advantageously eliminate or reducecross-coupling between the two or more sensors 101, 102, as illustratedin FIGS. 14A and 14B. For the case of FIGS. 14A and 14B, the backchamberlength is increased to 23 mm, such that its Helmholtz resonance becomes18 kHz. In this way, the sensor resonances are very close to thebackchamber Helmholtz resonance, and cross-coupling between the firstsensor 101 and the second sensor 102 is substantially eliminated. Incertain embodiments, the Helmholtz resonance of the backchamber 65 andthe sensor resonance are less than 1%, less than 2%, less than 3%, lessthan 5%, less than 8%, or less than 10% from each other. While theresponse curves of FIGS. 14A and 14B are plotted for the case when thereis a first sensor 101 and a second sensor 102 in parallel, the curvessubstantially match the curve for an individual sensor, with no othersensor parallel to it. Thus, optimizing the Helmholtz resonance incertain embodiments can be an effective way to reduce or completelyeliminate cross-coupling.

Air Bubbles to Increase Sensitivity

For a sensor 10 generally assembled in air, when it is immersed intowater, water will gradually fill the sensor 10, which can provideinsensitivity to hydrostatic pressure. Sometimes, however, some amountof air may remain inside the sensor 10 and one or more gas or airbubbles (ranging in size between about 0.1 mm and about 2 mm diameter)can be trapped inside the sensor 10. It is possible to generally avoidsuch gas or air bubbles by putting a surfactant into the water, such asa standard dish soap detergent, so that the surface tension of water isreduced, and water can flow easily into the sensor 10. In certainembodiments, however, it is beneficial to keep the one or more gas orair bubbles inside the sensor 10, or to introduce one or more gas or airbubbles deliberately into the sensor 10. In certain embodiments, the oneor more gas or air bubbles advantageously generally increase thesensitivity of the sensor 10, while reducing its frequency bandwidth.

For example, in certain embodiments, the presence of a small air bubblein the backchamber 65 has a negligible effect on the acoustic mass.However, because of the compressibility of water is very small, thestiffness of the backchamber 65 can be dominated by the compressibilityof the air bubble. The overall stiffness of the diaphragm 20 andbackchamber 65 system can therefore be reduced in certain embodiments,which decreases the resonance frequency. The reduction in resonancefrequency in certain embodiments is not strongly dependent on the sizeof the air bubble (as long as it is larger than approximately 100 μm),since the mass is generally dominated by water, and the compressibilityis generally dominated by air. Certain embodiments of the sensor 10 canadvantageously measure pressures as low as 3.5 μPa/Hz^(1/2) in afrequency range of 100 Hz to 10 kHz. This enhanced minimum detectablepressure can be provided by the increased compressibility in thebackchamber 65 caused by the trapped air. In certain embodiments, thesensor 10 can advantageously measure pressures less than 10μPa/Hz^(1/2), less than 9 μPa/Hz^(1/2), less than 8 μPa/Hz^(1/2), lessthan 7 μPa/Hz^(1/2), less than 6 μPa/Hz^(1/2), less than 5 μPa/Hz^(1/2),less than 4 μPa/Hz^(1/2), or less than 3 μPa/Hz^(1/2).

Thus, in certain embodiments, the one or more gas or air bubbles may beused where sensitivity is more significant for the application of thesensor 10, and bandwidth can be sacrificed. The one or more gas or airbubbles serve as a generally compressible (e.g., more compressible thanwater) and generally elastic element within the sensor 10 whichsubstantially dominates the compressibility of the contents of thesensor 10.

Fabrication Process

In certain embodiments, the fabrication process of the acoustic sensor10 involves silicon microfabrication techniques. FIG. 15 schematicallyillustrates an example fabrication process in accordance with certainembodiments described herein. Other techniques are possible. Asilicon-on-insulator (SOI) wafer includes a silicon substrate 510, aburied oxide layer 520 having a thickness of approximately 1 μm, and asilicon device layer 530 having a thickness of approximately 450 nm. Alow temperature oxide (LTO) layer 540 is deposited on the SOI wafer, asshown in (a) of FIG. 15. Then, the wafer is coated with photoresist 550and exposed using photolithography, e.g. using a photolithography mask560, as shown in (b) of FIG. 15. The LTO layer 540 is then etched withplasma etch to form the structure shown in (c) of FIG. 15. Thispatterned LTO layer 540 is used as a hard mask to etch the silicon layer530 underneath, as shown in (d) of FIG. 15. Once the front side ispatterned with the photonic-crystal structure, as shown in (e) of FIG.15, the back side is patterned to release the photonic-crystal structureof the silicon device layer 530.

FIG. 16 schematically illustrates an example fabrication process forproducing a backside pattern in accordance with certain embodimentsdescribed herein. As shown in (a) of FIG. 16, a low temperature siliconoxide (LTO) layer 540 is deposited on the silicon substrate 510 (e.g.,the silicon substrate 510 resulting from the process described above inconjunction with FIG. 15). As drawn in (a) of FIG. 16, a LTO layer 540can be deposited on each of both sides of the silicon substrate. Incertain embodiments, one or more nitride layers (e.g., Si₃N₄ not shown)can be deposited on each of the LTO layers 540. These nitride layers canhelp compensate for residual stresses in the silicon layer 530. Forexample, in certain embodiments, Si₃N₄ is deposited under tensilestress, which can compensate for the compressive stress due to thesilica (SiO₂) films. The LTO layer 540 on the surface of the siliconsubstrate 510 opposite to the silicon device layer 530 is patternedusing reactive ion etching as shown in (b) of FIG. 16. At least aportion of the silicon substrate 510 on the backside is removed (e.g.,using tetramethylammonium hydroxide (TMAH) wet etch), as shown in (c) ofFIG. 16. Finally, as shown in (d) of FIG. 16, at least a portion of theburied oxide layer 520 and the remaining portions of the LTO layers 540on each side of the silicon substrate 510 are removed (e.g., usinghydrofluoric-acid) to release the structure of the silicon device layer530. With this example fabrication method, more than 250 chips can befabricated on a 4-inch wafer. Utilizing the parallel fabrication processprovided by the example photolithography process of FIGS. 15 and 16, thenumber of sensors 10 that can be fabricated at a given time can beincreased substantially, thus the cost can be reduced, which can be veryimportant for commercial mass production.

In certain embodiments, as shown in FIG. 1B, the body of the sensor 10,e.g. the structural element 50 can be fabricated of a plurality ofelements. For example, the structural element 50 can be fabricated bybonding together several wafer portions (e.g., portions of 4″-diameterfused silica wafers), each of which has a different pattern of holes,e.g., pressure equalization channels as discussed above. FIGS. 17A-17Cschematically illustrate example portions of three individual wafers 50a, 50 b, 50 c and their patterns of holes to be used as building blocksof the silica structural element 50 in accordance with certainembodiments described herein. While the holes of FIGS. 17A-17C aregenerally circular, other shapes of holes (e.g., square, rectangular,triangular, polygonal, oval, or irregular) may also be used. Thediameters of the example holes are 0.3 mm, 2 mm, and 0.2 mm for 50 a, 50b, and 50 c of FIGS. 17A-17C, respectively. The diameters of the holescan be tailored to other diameters. FIG. 18 schematically illustratesexample locations of the wafers 50 a, 50 b, 50 c of FIGS. 17A-17C beingbonded together and attached to the photonic-crystal structure of thediaphragm 20 and the optical fiber 30 to form the sensor head inaccordance with certain embodiments described herein. In thisembodiment, the photonic-crystal structure serves as the reflectiveelement 22 of the diaphragm 20. The wafer thicknesses in certainembodiments are 0.5 mm. In other embodiments, the wafer thicknesses canbe between about 0.3 mm and 0.7 mm, or between about 0.4 mm and 0.6 mm.Both sides of each wafer (e.g., 50 a, 50 b, 50 c of FIGS. 17A-17C) canbe polished for bonding purposes.

For producing the wafer portions, a two-dimensional array of circularholes can be etched through each wafer with the pattern or arraycomprising a plurality of cells with each cell corresponding to onesensor head. For example, FIGS. 17A-17C only show one cell of thispattern for the three wafers 50 a, 50 b, 50 c with the one cell utilizedto form one sensor head. In certain embodiments, the hole closest to thediaphragm 20 (e.g., shown in FIG. 17A and having a diameter of 0.3 mm)defines the dimension over which the diaphragm 20 of the acoustic sensor10 will be allowed to flex, which affects the acoustic sensitivity ofthe final device (e.g., the larger the diaphragm 20, the more sensitivethe sensor 10). The second and third layers (e.g., shown in FIG. 17B andFIG. 17C) of certain embodiments define the channels for the water flowfrom the diaphragm 20 to the backchamber 65 shown in FIG. 18 (e.g., inthe case of a hydrophone). In certain embodiments, silica wafersproduced at Valley Design of Santa Cruz, Calif. and patterned by MindrumPrecision of Rancho Cucamonga, Calif. can be utilized.

In certain embodiments, following the fabrication of thephotonic-crystal structure of the diaphragm 20, the silicon-on-insulator(SOI) wafer is bonded to the silica wafers using a technique calledsilicate bonding (hydroxide-catalysis bonding as described in the LaserInterferometer Gravitational-Wave Observatory (LIGO) project). In thismethod, a hydroxide catalyzes the silica surface by hydration anddehydration. Because the surfaces are desired to be in close contact tobond, a flatness of λ/10 or better is used on the surfaces in certainembodiments. Furthermore, in certain embodiments, hydrophilic surfaceswith a high density of Si—OH groups are utilized for a successfulbonding. The procedure applied to achieve the bonding in certainembodiments includes rinsing the substrates under de-ionized (DI) waterto wash off any particles, and wiping the surface with methanol to dry.Next, approximately 5 ml from a sodium silicate solution are drawn witha pipette, and DI water is transferred to the sodium silicate solutionto obtain approximately 25 ml (1:4) of bonding solution. Approximately1.0 ml of this bonding solution is extracted using a fresh pipette, anddispensed onto the glass. Then, the two surfaces to be bonded arebrought together into contact with pressure.

In certain embodiments, this process is utilized to bond the SOI wafershaving the diaphragms to the silica wafers, each of which are againbonded to another silica wafer using the same silicate bondingtechnique. In certain embodiments, two silica wafers, e.g., 50 b shownin FIGS. 17B and 50 c shown in FIG. 17C, are bonded together. On top ofthis stack, another silica wafer, e.g., 50 a shown in FIG. 17A, isbonded to the wafer 50 b. Then, the SOI wafer is bonded to wafer 50 a.The alignments utilized during this example process to center thecorresponding holes can be performed under a microscope. This stackcomprising four 4″ wafers, and the 2D array of sensors can be diced intoindividual cells to obtain the individual sensors. Bonding can be doneusing silicate bonding, or thermal bonding, as examples.

Once the silica wafers 50 a, 50 b, 50 c and the SOI wafer comprising thediaphragm 20 are bonded using silicate bonding technique, the sensor 10is further assembled. The sensor 10 assembly process of certainembodiments comprises holding the sensor head fixed with a vacuum chuckand moving an optical fiber 30 with a reflective element 32 (e.g., atthe tip of the optical fiber 30) in close proximity to the diaphragm 20.During this process, the reflection spectrum can be monitored, with thecavity length 40 inferred from a classic measurement of the Fabry-Perotcavity free spectral range. Once the correct cavity length 40 isachieved, the optical fiber 30 can be bonded to the structural element50. In certain embodiments, the fabrication process can be used to bondentire wafers together, then to dice into individual structural elements50, as described above. Alternatively, in certain embodiments, thewafers can be diced first, then bonded into individual structuralelements 50 one at a time. In certain embodiments, the optical fiber 30can be mounted after dicing the wafers. In other embodiments, theoptical fiber 30 can be mounted before dicing the wafers.

In certain embodiments, the method used to bond the optical fiber 30 tothe structural element 50 advantageously provides a Fabry-Perot cavity40 with a reproducible cavity length, e.g., a resonance wavelength thatsubstantially does not change during the bonding process, during thecuring process if the bonding requires curing, and over time after thedevice assembly is completed. In certain embodiments, it alsoadvantageously yields a bond that substantially does not produce achange in the cavity length 40 as temperature varies. This goal can bemet using a number of techniques, e.g., phenyl benzoate, arc splicing,or CO₂-laser fusion.

In certain embodiments, as shown in FIG. 19, for the phenyl benzoateapproach and for the epoxy approach, two through holes 85 of diameters0.75 mm are drilled symmetrically on the sides of a silica capillarytube 80 of internal diameter close to that of the optical fiber 30 (forexample, a diameter of 127 μm for a standard fiber, which typically havea diameter of 125 μm). The outside diameter of the capillary tube 80 isnot critical; a value of 1.8 mm can be used. The holes 85 providechannels for the bonding material to reach and hold the optical fiber 30inside the capillary tube 80. Phenyl benzoate (C₁₃H₁₀O₂), which is apowder at room temperature, is a bonding material which is compatiblewith certain embodiments described herein. The optical fiber 30 ofcertain embodiments is inserted into the capillary tube 80 to form aFabry-Perot cavity 40 with the reflective element 20 of the diaphragm20. The optical spectrum of the sensor 10 is monitored with an opticalspectrum analyzer while the fiber end is brought in close proximity tothe diaphragm 20 by a high-accuracy mechanical positioner. Once thecorrect cavity length is achieved, phenyl benzoate is applied throughthe holes 85, then heated above the melting point of phenyl benzoate(phenyl benzoate melts at 68° C.-70° C.). The molten phenyl benzoateflows into the holes 85, the heat source is removed, and the phenylbenzoate cools down and crystallizes as it solidifies and bonds theoptical fiber 30 to the capillary tube 80. After the bonding process iscomplete, the output spectrum can be reexamined in certain embodiments.If a deviation is observed in the Fabry-Perot cavity length from thetarget value due to the bonding process, the phenyl benzoate can bereheated, at which point the optical fiber 30 is free to move, thecavity length is adjusted, and the spacing is measured again. Theprocess can be repeated a few times until the desired cavity length isachieved. One further advantage of this example method is that becausethe side holes 85 are symmetrically located, the bonding material exertsequal forces on the optical fiber 30 (see FIG. 19), which wouldotherwise result in a shift in the cavity length.

In another example fabrication method, an electric arc (for example froma commercial fiber splicer) is used to attach the optical fiber 30 tothe silica capillary tube 80. FIGS. 20A-20B schematically illustratestructures used in a method that reduces the arc current used to obtaina good bond between the two elements in accordance with certainembodiments described herein. The capillary tube 80 of certain suchembodiments is tapered at one end 81. The optical fiber 30 is insertedin the capillary tube 80, and the whole assembly is placed in aconventional arc splicer. With suitable choice of the arc current andarc duration, the optical fiber 30 is fused to the silica capillary tube80. A similar goal can be met using a CO₂ laser. As illustrated in FIGS.20A-20B, the untapered, large area side 82 of the silica capillary tube80 is bonded to the side of the silica structural element 50 where thediaphragm 20 does not exist (e.g., side 82 shown in FIG. 20A is bondedto side 51 of the structural element 50 shown in FIG. 20B, which is notto scale). In certain embodiments, this bond can be achieved by silicatebonding, which works successfully for silica surface bonding purposes asdescribed previously.

FIG. 21 is a flowchart of an example method 1000 of fabricating anacoustic sensor 10 in accordance with certain embodiments describedherein. The method 1000 comprises providing a diaphragm 20 comprising areflective element 22, as shown in operational block 1010 of FIG. 21.The method 1000 also comprises positioning an optical fiber 30 relativeto the reflective element 22 such that light emits from the opticalfiber 30 and is reflected from the reflective element 22, as shown inoperational block 1020. Positioning the optical fiber 30 relative to thereflective element 22 comprises forming an optical cavity 40therebetween. The method 1000 further comprises, as shown in operationalblock 1030, mechanically coupling the diaphragm 20 to the optical fiber30 with a structural element 50. The structural element 50 comprisessilica.

In certain embodiments of the method 1000, providing a diaphragm 20comprising a reflective element 22, as shown in operational block 1010,comprises providing a photonic-crystal structure as the reflectiveelement 22. Providing a photonic-crystal structure can compriseproviding a photonic-crystal structure fabricated by photolithography.In certain embodiments, the method 1000 further comprises silicatebonding the diaphragm 20 to the structural element 50.

In various embodiments, the method 1000 further comprises employing anelement 25 comprising silica within the optical cavity 40. The method1000 can further comprise selecting a thickness for the element 25comprising silica approximately equal to a distance between the firstend 32 of the optical fiber 30 and the diaphragm 20.

In certain embodiments, the method 1000 can further comprise selecting adiaphragm diameter to increase mechanical compliance and/or selecting adiaphragm cross-sectional area to increase mechanical compliance. Themethod 1000 can also include employing one or more fluid conduits 55separate from the reflective element 22 positioned on the diaphragm 20.The method 1000 can also include employing at least one generallycompressible and generally elastic element within the optical cavity 40to increase sensitivity.

Example Embodiment and Characterization of an Example Optical AcousticSensor

FIG. 22 schematically illustrates an example acoustic sensor 10fabricated and assembled in accordance with certain embodimentsdescribed herein. In this example, the sensor 10 comprises a deflectablediaphragm 20 comprising a 450 nm thick single-crystal siliconphotonic-crystal structure placed in close proximity (approximately 25μm) to the stationary tip 32 of a single-mode fiber 30 coated with 15 nmof gold serving as the reflective element 32 of the optical fiber 30.The photonic-crystal structure is a square 300 μm on each side. Thephotonic-crystal structure has a square lattice of 800 nm diameter holeson a 900 nm pitch, e.g., a minimum wall thickness on the order of 100nm. The sensor 10 was fabricated using the photolithography and silicatebonding methods described herein. FIG. 23A shows a scanning electronmicrograph of a top view of the fabricated photonic-crystal structure ofthe diaphragm 20. FIG. 23B shows a scanning electron micrograph of anangle view of the fabricated photonic-crystal structure of the diaphragm20. Shown in FIG. 23C, the fabricated sensor 10 is 5×5×5 mm indimension. Further miniaturization is possible.

Experimental Characterization of the Example Optical Acoustic Sensor

The example fiber acoustic sensor 10 was tested in an acousticallyisolated enclosure using a conventional calibrated microphone as areference. A schematic of the acoustic characterization setup is shownin FIG. 24. The sensor 10 was interrogated with a 15-kHz linewidth,low-noise, 1550-nm laser diode. The laser light was coupled into thesensor 10 via an optical circulator, and the light reflected by thesensor 10 was detected with a PIN photodiode.

The electrical outputs of the calibrated reference microphone and theacoustic sensor 10 were connected to a dynamic signal analyzer (DSA),which measured the frequency response and noise spectrum of the twosensors, and the coherence between them. Coherence is a measure of thedegree of linear correlation between two signals. When two signals areuncorrelated, such as if one is dominated by noise, the coherence valueis zero. In the case of complete correlation, the coherence value isone. The electrical signal from the reference microphone was used as afeedback signal on the DSA to adjust the output of the acoustic sourcesuch that a constant pressure of 1 Pa was incident on the sensors at allfrequencies. The measurements were conducted up to an acoustic frequencyof 30 kHz, which is the frequency band that the calibrated referencemicrophone is specified and measured to have a flat frequency response.

FIG. 25 shows that there is a strong coherence (˜1) between the twosensor outputs for frequencies higher than ˜700 Hz, which establishesthat the data in this frequency band is accurate. The low coherence atlower frequency might be because one or both of the sensors has asignal-to-noise lower than unity. FIG. 26 shows the measured frequencyresponse of the example fiber acoustic sensor 10, which is the ratio ofthe power spectrum of the acoustic fiber sensor 10 (in volts V) and thepower spectrum of the reference microphone (in Pascals Pa). This sensor10 has a measured finesse of ˜6. There is a flat band with a bandwidthgreater than 8 kHz, and a resonance arising from a mechanical resonancein the FP sensor at ˜12.5 kHz. The other small resonances above ˜700 Hzare mainly due to residual resonances in the acoustic chamber.

FIG. 27 plots three example noise curves, measured on a differentexample sensor 10 with the same apparatus as FIG. 24 but in the absenceof acoustic signal. In this example, noise can be measured between about10⁻⁴ and about 10⁻⁷ V/Hz^(1/2). The top curve is the noise of the sensor10. The middle curve is the noise measured when replacing the acousticfiber sensor 10 with a reflector. The lowest curve was measured when thelaser was off. It represents the noise due to the detection electronics(photodetector plus DSA). The middle curve represents the optoelectronicnoise: it includes the detection electronics and the laser noise. It hasa white-noise component dominated by the relative intensity noise of thelaser (−141 dB V/√Hz) at the highest frequencies, and by a 1/f noisecomponent below ˜10 kHz. The sensor 10 noise (top curve) is a littlelarger than the optoelectronic noise; this increase can be assigned tothermo-mechanical noise of the sensor 10, as well as conversion of laserphase noise into intensity noise.

FIG. 28 shows the minimum detectable pressure (MDP) of the sensor 10with the frequency response shown in FIG. 26. This spectrum was obtainedby dividing the noise spectrum of this sensor 10 by its frequencyresponse. The data below ˜700 Hz is again not accurate because of thelow coherence of the measurement. From ˜700 Hz to 8.6 kHz, the MDP is inthe range of ˜180 to ˜27 μPa/√Hz. Above 8.6 kHz the MDP improves as thefrequency approaches the device's main mechanical resonances, andbecomes as low as about 5.6 μPa/√Hz at 12.5 kHz. The average MDP overthe frequency band of 1 kHz to 30 kHz is about 33 μPa/√Hz. This resultdemonstrates that certain embodiments as described herein are capable ofproviding the sensitivity and bandwidth performance desired forimplementation in, e.g., Navy acoustic systems. In certain embodiments,these MDP values are limited in part by the optoelectronic noise, andthey can be further reduced with straightforward improvements in thedetection electronics. At low frequencies, the acoustic noise of thelaboratory environment can also be responsible for some degradation inthe MDP observed during measurements.

Thermal Stability of the Example Optical Acoustic Sensor

To characterize the thermal stability of the example fiber acousticsensor 10, its temperature was varied from 2° C. to 58° C., and theresulting shift in the resonance wavelengths was recorded using anoptical spectrum analyzer. The same measurement was also carried outwith a first-generation fiber sensor, e.g., silica fiber and siliconstructural element. The variations in resonance wavelength measured forthe two sensors are shown in FIG. 29. The normalized wavelength shiftwith temperature is ˜70 times lower in the sensor of certain embodimentsdescribed herein than the earlier sensor (3.3×10⁻⁶/° C. v. 2.4×10⁻⁴/°C.). The thermal stability of the all-silica fiber sensor (e.g., silicafiber 30, silica capillary tube, 80 and silica structural element 50)was expected to be Δλ/λ/ΔT=α_(SiO) ₂ =0.55×10⁻⁶/° C. The measured valueis higher mostly because of the silicate-bonding material. For certainembodiments, this result constitutes a major step towards highly stableacoustic sensors, and it is more than adequate for larger-scale sensornetworks.

Example Embodiment and Characterization of an Example Optical AcousticSensor System

FIG. 30 is an example optical acoustic sensor system 200 for oceanacoustics. See also 0. Kilic, M. Digonnet, G. Kino, and O. Solgaard,“Photonic-crystal-diaphragm-based fiber-tip hydrophone optimized forocean acoustics,” Proc. of SPIE, vol. 7004, 700405 (2008). In theexploded view of FIG. 30, the sensor system 200 comprises a face 210 ofthe sensor head. The face 210 of the sensor head can comprise areflective element 22 of a diaphragm 20 and a structural element 50 inaccordance with certain embodiments described above. The sensor system200 in this embodiment can comprise a plurality 230 of optical fibers30, each having a reflective element 32 on the end of the optical fiber30. The sensor 200 can further comprise a backchamber housing 260. Across-section of the sensor system 200 is similar to the sensor system100 shown in FIG. 12.

Referring to FIG. 12, the reflective element 20 can comprise aphotonic-crystal reflective element 22 micro-machined on a siliconstructural element 50. As shown in FIG. 30, the plurality 230 of opticalfibers 30 can comprise four SMF-28 fibers, each one transmitting andreturning a different optical signal. In other embodiments, more thanfour optical fibers 30 can be used. As depicted in FIG. 12, three of thefour optical fibers 30 can lead to the photonic-crystal reflectiveelements 22 of the diaphragm 20 placed at the face of the sensor head.In this embodiment, the photonic-crystal structures are reflectiveelements 22 with high reflectivity (>95%). In accordance with certainembodiments described herein, the ends of each optical fiber 30 can becoated with a reflective element 32, so that when placed in closeproximity (˜20 μm) with the photonic-crystal reflective elements 22,they each form a Fabry-Perot optical cavity 40. By deforming thecompliant diaphragm 20, an incident acoustic signal 400 can modulate thespacings of the optical cavities 40, giving rise to a change in thepower of the laser light reflected back into the optical fibers 30.

The three sensors of the sensor system 200 shown in FIG. 30 can belocalized in a region of about 2.5-mm diameter which is approximately anorder of magnitude smaller than one of the shortest acoustic wavelengthsof interest (15 mm at 100 kHz), so they can be exposed to approximatelythe same acoustic amplitude. In other embodiments, the sensor system 200can be localized in a region of about 2 mm to about 3 mm in diameter.The three diaphragms 20 can have different diameters (e.g., 150 μm, 212μm, and 300 μm) and hence different compliances (relative compliancesare, e.g., ×1, ×4, and ×16, respectively). In other embodiments, thediaphragms 20 can have diameters between about 100 μm and about 400 μm,between about 150 μm and about 350 μm, or between about 200 μm and about300 μm. As disclosed herein, each of the three sensors can address adifferent range of pressures to increase the dynamic range of the sensor200 head over that of a single sensor. Calculations show that this rangecan span pressures as low as the ocean's ambient thermal noise (˜10μPa/Hz^(1/2)) all the way up to 160 dB larger signals. The fourth fibercan be connected to a reference reflective element for calibrationpurposes in sensor-array applications. It can provide a static referencesignal transmitted along with the acoustic signals from the other threefibers in order to account for loss and noise associated with the paththrough which the signals travel. As described herein and shown in FIG.12, the sensor system 200 can comprise diaphragm-sized channels 90,expanded channels 92, and a backchamber channel 95.

Fabrication of the Example Optical Acoustic Sensor System

The sensor system 200 can be fabricated using silicon micro-fabricationtechniques, for example, as shown in FIG. 31. The fabrication caninclude the following steps: (a)-(c) etching the diaphragm-sizedchannels 90 and expanded channels 92, (d) bonding the backside wafer,(d) defining the photonic-crystal reflective elements 22, and (e)etching the backchamber channels 95 and fiber alignment channels 97.

In (a) of FIG. 31, a 2-μm-thick low-temperature silicon oxide (LTO)layer 610 is deposited using low-pressure chemical-vapor deposition(LPCVD) on both sides of a 400-μm-thick silicon wafer 620. In otherembodiments the wafer 620 can be between about 300-μm-thick and about700-μm-thick, between about 350-μm-thick and about 650-μm-thick, orbetween about 400-μm-thick and about 600-μm-thick. LTO, instead ofthermally grown silicon-oxide, has a low stress, which can beadvantageous in later fabrication steps. The LTO layer 610 on thebackside is subsequently patterned using a wet oxide etch (bufferedhydrofluoric acid). This step defines the shapes of the expandedchannels 92.

Before etching the expanded channels 92 into the wafer 620, the backsideis covered with a thick (>10 μm) photoresist (not shown) and ispatterned with shapes defining the diaphragm-size channels 90 as shownin (b) of FIG. 31. This step also determines the diameters of each ofthe three diaphragms 20. The next step involves etching the backsideusing deep-reactive ion-etching (DRIE). The etch can be timed so thatthe diaphragm-size channels 90 are only etched partially into the wafer620.

In (c) of FIG. 31, the resist is then stripped off, so that the LTOlayer 610 on the backside with the expanded channel patterns areexposed. Then, a second DRIE step is employed, so that the expandedchannels 92 are etched partially into the wafer 620. This step alsocontinues to etch the partially etched diaphragm-size channels 90, untilthey reached the LTO layer 610 on the frontside.

In (d) of FIG. 31, the LTO layer 610 on the backside is stripped offusing a timed hydrofluoric-acid etch, while protecting the front side ofthe wafer 620 with photoresist. Afterwards, a second 400 μm-thicksilicon wafer 620 is bonded to the backside of the first wafer 620 at1000° C. In other embodiments the second wafer 620 can be between about300-μm-thick and about 700-μm-thick, between about 350-μm-thick andabout 650-μm-thick, or between about 400-μm-thick and about600-μm-thick. This is followed by LPCVD deposition of the diaphragmlayer 630, comprising of a 450-nm polysilicon layer sandwiched betweentwo 25-nm silicon nitride layers. In other embodiments the totaldiaphragm can be between about 400-μm-thick and about 700-μm-thick, orbetween about 450-μm-thick and about 650-μm-thick. The thin nitridelayers can serve to compensate for the residual stress in thepolysilicon layer. See, e.g., S. Kim, S. Hadzialic, A. Sudbo, and O.Solgaard, “Single film broadband photonic crystal micro-mirror withlarge angular range and low polarization dependence,” in Conference onLasers and Electro-Optics (CLEO), Baltimore, Md., p. CThP7, (2007). Thislow stress can provide relatively flat diaphragms 20, e.g, as shown inFIG. 32.

FIG. 32 shows an optical profilometry measurement on a mid-sizeddiaphragm (e.g., 212 μm diameter) comprising of a 450-nm polysiliconlayer sandwiched between two 25-nm silicon nitride layers, without anyoxide underneath. The measurement indicates that the diaphragm 20 iselevated from the surface of the wafer 620 by a relatively small amountof about 300 nm. The top region is flat within about 10 nm in a 130-μmdiameter central region.

After the diaphragm layer 630 is deposited, it is patterned with aphotonic-crystal mirror pattern. To make the holes of thephotonic-crystal reflective elements 22, a relatively thin (e.g.,between about 100 nm and about 200 nm, here, about 150 nm) LTO layer 610is deposited to serve as an etch mask (not shown). Apolymethylmethacrylate (PMMA) resist layer is spun on the LTO layer 610and is patterned with e-beam lithography. Other method of patterning arepossible, e.g., photolithography. The holes of the photonic-crystalreflective elements 22 are then etched into the LTO layer 610 usingmagnetically-enhanced reactive-ion etching (MERIE), and then into thediaphragm layer 630 through subsequent MERIE etches.

In (e) of FIG. 31, DRIE is used to etch the backchamber channel 95 andthe fiber alignment channels 97 on the backside. During the etch, thephotonic-crystal reflective elements 22 are protected by the remainingLTO layer 610, which is stripped off in the last step with a shorthydrofluoric-acid etch.

The various channels are etched into the silicon wafer 620anisotropically through a series of alternating passivation andisotropic-etch steps, which can create scalloping on the sidewalls witha mean depth of ˜0.25 μm (or between about 0.15 μm and about 0.35 μm orbetween about 0.2 μm and about 0.3 μm). In the passivation step, aplasma conformally deposits a layer of a PTFE-like fluorocarbon polymer.This polymer protects the sidewalls from etching, and remains thereafter the etch is completed. The hydrophobic nature of this passivationfilm, combined with the scalloping geometry of the sidewalls, makes thewetting of the DRIE etched channels substantially low. In certainembodiments, this makes it challenging to properly fill the opticalcavity 40 with water. To provide sufficient wettability in certainembodiments, the sidewall polymers can be removed, in some embodimentscompletely removed, after the DRIE steps. Employing an asher that etchesorganics away with an oxygen plasma, followed by a hot Piranha wet etch(9:1 sulfuric acid:hydrogen peroxide) is sufficient in certainembodiments for stripping off the sidewall polymers in the wafers 620.

The reflective elements 32 on each of the optical fibers 30 within theplurality 230 of fibers 30 can be deposited using e-beam evaporation oncleaved SMF-28 fibers. The reflective elements 22 in the exampleembodiment comprised of a 4-nm chrome adhesion layer, followed by a20-nm gold reflection layer, and finally a 15-nm magnesium fluorideprotection layer. In other embodiments, the chrome layer can be betweenabout 2-nm and about 5-nm thick. The gold reflection layer can bebetween about 15-nm and about 25-nm or between about 18-nm and about22-nm thick. Additionally, the protection layer can be between about10-nm and about 20-nm or between about 18-nm and about 22-nm thick.Other dimensions are possible. Gold is advantageous because of its lowabsorption and superior reflective properties at wavelengths around 1550nm, the operation wavelength of the laser to address the sensor system200.

Methods of assembly of the plurality 230 of fibers 30 to the structuralelement 50 described herein in certain embodiments of sensors 10 can beemployed in sensor systems 200, The optical fibers 30 are pushed throughthe fiber alignment channels 97 while the spectrum of the Fabry-Perotinterferometers are monitored using an optical-spectrum analyzer. Afterthe target spacing for the Fabry-Perot optical cavity 40 is reached, theoptical fibers 230 are secured with epoxy (while monitoring andadjusting the spacing). Finally, the face 210 of the sensor head isattached to the backchamber housing 260 with epoxy, and the plurality230 of optical fibers 30 are tighted with heat-shrink tubing. Thebackchamber housing 260 can be a commercial ball-end hose barb. Acomplete, packaged sensor system 200 is shown in the photograph in FIG.33.

Theoretical Modeling

The optimization of a sensor system 200 design for ocean acoustics ischallenging in that the ocean noise spectrum is complex, and an analysisof the parameter space utilizes interdisciplinary modeling: opticalmodeling of the displacement detection, mechanical modeling of thediaphragm motion, acoustic modeling of the sensor baffle and thebackchamber design, and microfluidics modeling of the channelstructures. Also, a single parameter can affect several sensor featuressimultaneously. For example, the size of the perforations in thediaphragm affect the optical reflection, the hydrostatic sensitivity,and the mechanical compliance of the diaphragm. Hence, an optimizationprocess through a direct finite-element numerical simulation isimpractical, and it also does not provide insight into how the varioussensor parameters can be adjusted. Therefore, an analytical model asdescribed herein is utilized that provides information on how the designparameters can be tailored to meet the demands of ocean acoustics.

Lumped-Element Equivalent-Circuit Model

The characteristic sensor system dimensions (˜1 mm) are substantiallysmaller than the acoustic wavelengths of interest. In other embodiments,the sensor system dimensions can be between about 1.5 mm and about 2 mm.Therefore, it is possible to approximate spatially distributed elementswith a single lumped element to model the noise of the sensor system 200and thermal-mechanical noise. See, e.g., T. B. Gabrielson, “Mechanicalthermal noise in micromachined acoustic and vibration sensors,” IEEETrans. Electron Devices, vol. 40, pages 903-909 (1993). In this lumpedmodel, the distributed potential and kinetic energies in the sensorsystem 200 are described through a single acoustic compliance C andacoustic mass M, respectively. Likewise, the dissipation in the sensorsystem 200 is modeled with a single acoustic resistance R.

Using lumped elements to describe the physical mechanisms in the sensorsystem 200, it is possible to analyze the sensor system 200 through anequivalent circuit formed by these elements, as shown in FIG. 34. Inthis circuit, the acoustic compliance C is similar to electricalcapacitance, and the acoustic mass M is analogous to electricalinductance. In the same convention, the acoustic resistance R is similarto electrical resistance. The acoustic impedances (Z) for these lumpedelements are 1/(jωC), jωM, and R, respectively. The relationship betweenthe pressure drop (P) and flow rate (ū) across these impedances isassumed to be P=ūZ, which is valid as long as the flow is not turbulentor the diaphragm displacement is small compared to its thickness. Forclarity, a model of only one sensor 10 is shown in FIG. 34, while inthis example, there are three different sensors in parallel in thesensor system 200 connecting to the same backchamber 65.

The incident acoustic signal is represented by a pressure source(P_(in)). The acoustic signal can travel to the optical cavity 40through two pathways, either as a volume flow through the holes of thephotonic-crystal reflective elements 22 (the path M_(hole)-R_(hole)), orthrough a motion of the compliant diaphragm 20. Once the signal reachesthe optical cavity 40, it is transmitted through the diaphragm-sizedchannel 90 around the optical fiber 30 leading to the backchamber 65.The small volume of the optical cavity 40 makes its acoustic compliancelow, which means that the water is not compressed between the opticalfiber 30 and the diaphragm 20 but is forced to flow into the backchamber65. Without the backchamber 65, the motion of the diaphragm 20 would beinhibited by a stiff optical cavity 40, so that the response of thesensor 10 would drop by more than 80 dB in water compared to air. Sincethe quantity measured by the optics of the sensor 10 is only thediaphragm displacement, this equivalent-circuit model can be used tocalculate the fraction of incident pressure that drops across thediaphragm compliance to obtain the sensor response. Similarly, theamount of noise transferred to the diaphragm compliance from dissipativeelements can be calculated using this equivalent-circuit model to obtainthe thermal-mechanical noise limitation of the sensor system 200.

Acoustic Impedance of the Diaphragm

The equation of motion for the small transverse displacement u of astretched circular diaphragm 20 fixed around its periphery, withthickness h, radius a, and density ρ is:

$\begin{matrix}{{\left( {{h\; \rho \frac{\partial^{2}}{\partial t^{2}}} + {D{\nabla^{4}{- h}}\; \sigma \nabla^{2}}} \right)u} = {P\; ^{{j\omega}\; t}}} & (5)\end{matrix}$

See, e.g., S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates andShells (McGraw-Hill, New York, 1959); and M. Di Giovanni, Flat andCorrugated Diaphragm Design Handbook (Marcel Dekker, New York, 1982).Here σ is the residual stress and D is the flexural rigidity, defined as

${D = {\frac{1}{12}E\; {h^{3}/\left( {1 - v^{2}} \right)}}},$

with E being Young's modulus, and ν Poisson's ratio. The diaphragm sizeis small in comparison to the acoustic wavelength, so the incidentpressure is modeled as a plane wave with amplitude P and frequency ω.(Frequencies in units of Hz refer to f=ω/2π).

Equation (5) can be solved analytically to obtain expressions for theresonance frequencies and mode profiles. The bending profile for adiaphragm 20 with low residual stress (e.g., a²hσ<<D) can be expressedas:

u(r,t)=u ₀ e ^(jwt)(1−r ² /a ²)²  (6)

here u₀=c_(m)P is the center displacement amplitude, and c_(m)=a⁴/64D isthe mechanical compliance (the inverse of stiffness) of the diaphragm20. In water, the mechanical resonances of the diaphragm 20 can bedisregarded, since the impedance of water dominates the diaphragmmechanics. Therefore, Eq. (6) is assumed valid over the frequency rangeof interest. For large displacements (u₀>h/5), the tensile stress of thebending diaphragm 20 becomes significant so that the diaphragm 20becomes harder to deflect for a given pressure. The center displacementin this case can be calculated with:

$\begin{matrix}{u_{0} = {{c_{m}P} - {\frac{\left( {1 + v} \right)\left( {7 - v} \right)}{16\; h^{2}}u_{0}^{3}}}} & (7)\end{matrix}$

See, e.g., M. Di Giovanni, Flat and Corrugated Diaphragm Design Handbook(Marcel Dekker, New York, 1982). Equation (5) models a solid diaphragm,hence does not account for the effects of the holes of thephotonic-crystal reflective element 22 on the diaphragm's mechanicalproperties. The perforations make the elasticity of the diaphragm 20highly anisotropic, which complicates the mechanical modeling.Nonetheless, it is possible to approximate the structure as a homogenousdiaphragm by using modified elastic constants. The effective elasticconstants of the photonic-crystal reflective element 22 are found byequating the strain energy of a perforated diaphragm to the one of anequivalent solid diaphragm. See, e.g., M. Pedersen, W. Olthuis, and P.Bergveld, “On the mechanical behaviour of thin perforated plates andtheir application in silicon condenser microphones,” Sensors andActuators A, vol. 54, pages 499-504 (1996).

A perforated plate can be modeled as a solid isotropic plate withmodified elastic constants. The effective elastic constants are found byequating the strain energy of the two plates, yielding the followingmaterial constants:

v ′  E ′ 1 - v ′2 = v   E 1 - v 2  ( 1 - 1 / 2 ) ( 8 ) E ′ 1 - v ′  2 = E 1 - v 2  ( ( 1 - 1 / 2 ) + 1 / 2  1 / 2  ( 1 - 1 / 2 ) 2  (1 - v 2 ) ) ( 9 )

It is possible to solve Eqs. (8) and (9) together to calculate theeffective Young's modulus E′, and the effective Poisson's ratio ν′.Alternatively, Eq. (9) yields the effective flexural rigidity D′:

D′=D(1−

)(1−½

^(1/2))+O(ν²)  (10)

Ignoring second order terms in ν yields the expression of D′ in Eq.(11).

ρ′=ρ(1−

)

σ′=σ(1−

)^(1/2)

D′=D(1−

)(1−½

)  (11)

Here

=0.50 is the fill factor, defined as the ratio of the open area to thetotal area of the photonic-crystal reflective element 22. The total areaon which the photonic-crystal reflective element 22 is defined (radiusof a_(PC)=25 μm) is smaller than the diaphragm 20 (radius of a=150 μm).Therefore, the elastic coefficients are not constant throughout thediaphragm 20. Specifically, the flexural rigidity of the diaphragmD_(dia) varies with radial position, such that D_(dia)(r>a_(PC))=D andD_(dia)(r≦a_(PS))=D′. To employ the simple model in Eq. (5), thecomposite diaphragm is assumed equivalent to a uniform diaphragm with aneffective flexural rigidity D″ satisfying approximately∇²D_(dia)∇²≡D″∇⁴. Similarly, a single density ρ″ is employed. Tocalculate these effective elastic constants, it is possible to usefinite-element analysis or the superposition method. A finite-elementsimulation of a composite 300-μm-diameter diaphragm with a50-μm-diameter central region represented with the modified elasticconstants of Eq. (11) yields an effective flexural rigidity (D″=0.76D),and an effective density (ρ″=0.70ρ). The residual stress is negligiblein the fabricated structures. These values were obtained for the lineardisplacement regime. A simulation accounting for the nonlineardisplacement regime yielded the same results, in agreement with Eq. (7).

The acoustic mass of the diaphragm 20 is determined by calculating thekinetic energy (U_(k)) of the diaphragm 20, then equating it to anequivalent system including of a lumped mass (M_(dia)) with a singlespeed (ū), defined as υ=∫₀ ^(a)υ(r)2πrdr corresponding to the volumetricflow rate. The results are assumed to be time harmonic as e^(jwt), henceu(r)=jωu(r). The acoustic mass of the diaphragm 20 is calculated thenusing

$U_{k} = {\frac{1}{2}M_{dia}{\overset{\_}{u}}^{2}}$

as:

$\begin{matrix}{M_{dia} = \frac{9h\; \rho^{''}}{5\; \pi \; a^{2}}} & (12)\end{matrix}$

Similarly, the potential energy (U_(p)) in the diaphragm 20 iscalculated, and then related to an equivalent system with a lumpedspring constant (k_(dia)) and a single displacement (ū), defined as ū=·₀^(a)u(r)2πrdr, which is the volume displacement. The equivalent springconstant is calculated using

$U_{p} = {\frac{1}{2}k_{dia}{{\overset{\_}{u}}^{2}.}}$

The compliance of the diaphragm (C_(dia)) is the inverse of this springconstant, hence from C_(dia)=1/k_(dia), the acoustic compliance is:

$\begin{matrix}{C_{dia} = \frac{\pi \; a^{6}}{192\; D^{''}}} & (13)\end{matrix}$

The compliance of the diaphragm 20 is of particular importance, becauseit determines the displacement of the diaphragm 20 as a function ofpressure. Since the optical part of the sensor 10 only senses thediaphragm displacement, the main purpose of the lumped model is tocalculate the pressure (P_(dia)) and noise across this particularcompliance.

Radiation Impedance of the Diaphragm

The ambient fluid plays an important role in the overall mechanics ofthe sensor 10, and necessitates modeling of other acoustic masses andcompliances that have a significant effect on the sensor dynamics. Thepresence of the fluid also creates dissipation, causingthermal-mechanical noise, which also utilizes modeling the loss throughan acoustic resistance. When calculating the acoustic mass andresistance, it can be assumed that in certain embodiments, the flow islaminar and the fluid is incompressible. To calculate the compliance,the compressibility of the fluid is taken into account.

In certain embodiments, the effective acoustic mass of the diaphragm 20in water is more than one order of magnitude larger than the acousticmass in vacuum. This is because the fluid moves with the diaphragm 20when it oscillates. Therefore, a mass term can be included to accountfor the moving fluid, referred to as the radiation mass (M_(rad)). Theoscillating diaphragm 20 also radiates part of its energy into thefluid, creating a channel of dissipation. To account for this radiativeloss, an acoustic radiation resistance (R_(rad)) can be included. Theradiation mass and resistance can be calculated by approximating thediaphragm as a rigid piston mounted in an infinite baffle, yielding:

$\begin{matrix}{M_{rad} = \frac{8\; \rho_{0}}{3\pi^{2}a}} & (14) \\{R_{rad} = {\frac{\rho_{0}}{2\pi \; c}\omega^{2}}} & (15)\end{matrix}$

See, e.g., L. L. Beranek, Acoustics (American Institute of Physics, NewYork, 1986); and M. Rossi, Acoustics and Electroacoustics (Artech House,Inc., 1988). Here ρ₀ is the density of the fluid and c denotes the speedof sound in the fluid. The modeling described herein uses the conventionof a frequency-dependent resistance in series with the mass reactance,in contrast to a constant shunt resistance parallel to the massreactance. See, e.g., L. L. Beranek, Acoustics (American Institute ofPhysics, New York, 1986).

In certain embodiments, an infinite baffle approximation is toosimplistic, considering that the sensor-head size is sub-wavelength overmost of the frequency range of interest. Since the sensor desirably hasa self noise that can be limited by radiation loss above 30 kHz, wherethe ocean noise is dominated by the Brownian motion of water molecules,the accurate modeling of the radiation loss can be significant incertain embodiments. A finite closed baffle may be a better descriptionof the structure in certain embodiments. Modeling a finite baffle can berather challenging, but the results can be summarized as follows: At lowfrequencies, the sensor acts like a piston at the end of an infinitetube, such that the radiation loss is approximately half of the valuefor an infinite baffle. At higher frequencies, when the size of the headbecomes comparable to the wavelength, the impedance values approachthose for an infinite baffle. See, e.g., P. H. G. Crane, “Method for thecalculation of the acoustic radiation impedance of unbaffled andpartially baffled piston sources,” J. Sound Vib., vol. 5, pages 257-277(1967); and T. Mellow and L. Karkkainen, “On the sound field of anoscillating disk in a finite open and closed circular baffle,” J.Acoust. Soc. Am., vol. 118, pages 1311-1325 (2005).

However, in characterization experiments and envisioned practicalapplications, the sensor 10 is mounted on a larger structure. Thetheoretical treatment based on the size, shape, and rigidity of suchactual baffle structures can be too complicated. Nonetheless, based onthe fact that these baffles are usually larger than the wavelengthsabove 30 kHz (<5 cm), the infinite baffle model in Eqs. (14) and (15)can be assumed sufficient in the modeling of certain embodiments of thesensor 10. If a more elaborate baffle model were to be used, the thermalnoise contribution to the ambient sea noise can be adjusted to reflectthe minimum noise level such a sensor-baffle structure is exposed to.

Flow Through Holes of the Photonic-Crystal Reflective Element

Water flowing through the holes of the photonic-crystal reflectiveelement 22 can encounter viscous resistance. The hole resistance has twocontributions, which are due to the horizontal flow of the fluid fromthe surroundings of the hole (squeeze-film flow), and the vertical flowof the fluid through the hole (Poiseuille flow). The horizontal-flowcontribution from each hole is:

R hole ↔ = 6   μ π   l 3  ( - 1 4  2 - 1 2  ln   - 3 4 ) , ( 16)

where m is the dynamic viscosity of the fluid and l is the cavityspacing. See, e.g., D. Homentcovschi and R. N. Miles, “Modeling ofviscous damping of perforated planar microstructures. Applications inacoustics,” J. Acoust. Soc. Am., vol. 116, pages 2939-2947 (2004); andZ. {hacek over (S)}kvor, “On acoustical resistance due to viscous lossesin the air gap of electrostatic transducers,” Acustica, vol. 19, pages295-299 (1967). In contrast to most microphones that employ a perforatedbackplate, the boundary conditions can prevent the diaphragm motion toinduce this squeeze-film flow. The perforated diaphragm 20 can be movedby the same pressure field that forces the flow through the holes. As aresult, the presence of the holes on the diaphragm 20 may notsignificantly reduce the squeeze-film damping.

The vertical-flow contribution from each hole, on the other hand, is:

$\begin{matrix}{R_{hole}^{\updownarrow} = \frac{8\; \mu \; h^{\prime}}{\pi \; a_{hole}^{4}}} & (17)\end{matrix}$

See, e.g., L. L. Beranek, Acoustics (American Institute of Physics, NewYork, 1986); M. Rossi, Acoustics and Electroacoustics (Artech House,Inc., 1988); and D. Homentcovschi and R. N. Miles, “Modeling of viscousdamping of perforated planar microstructures. Applications inacoustics,” J. Acoust. Soc. Am., vol. 116, pages 2939-2947 (2004). Inthe equation, an effective thickness

$h^{\prime} = {h + {\frac{3\pi}{8}a_{hole}}}$

can be employed. This modified thickness can be used to make correctionsfor the effect of the hole end, when the hole radius a_(hole) and thethickness h are comparable. See, e.g., D. Homentcovschi and R. N. Miles,“Viscous damping of perforated planar micromechanical structures,”Sensors and Actuators A, vol. 119, pages 544-552 (2005). The radiationresistance of the holes can be insignificant compared to the flowresistance and is not included in the modeling. The acoustic mass of thehole can also be considered, and taken as:

$\begin{matrix}{M_{hole} = \frac{4\; \rho_{0}h^{''}}{3\; \pi \; a_{hole}^{2}}} & (18)\end{matrix}$

See, e.g., L. L. Beranek, Acoustics (American Institute of Physics, NewYork, 1986); and M. Rossi, Acoustics and Electroacoustics (Artech House,Inc., 1988). To include the radiation mass of the holes, an effectivethickness

$h^{''} = {h + {\frac{2}{\pi}a_{hole}}}$

can be defined. Since the holes in certain embodiments provide parallelchannels, the overall hole impedance can be reduced by a factor equal tothe hole number.

Cavity Effects

The fluid moving through the optical cavity 40 to the diaphragm-sizedchannel 90 can encounter a resistance, referred to as squeeze-filmresistance:

$\begin{matrix}{R_{gap} = \frac{3\; \mu}{2\; \pi \; l^{3}}} & (19)\end{matrix}$

See, e.g., J. B. Starr, “Squeeze-film damping in solid-stateaccelerometers,” in IEEE Workshop in Solid-State Sensor and Actuator 4thTechnical Digest, pages 44-47 (1990). All the volume flow through theholes of the photonic-crystal reflective element 22 can go through theoptical cavity 40, hence its resistance can be expressed through Eq.(19). However, since the diaphragm diameter is significantly larger thanthe fiber diameter in certain embodiments, only a portion of the volumeflow induced by the moving diaphragm 20 has to flow through the cavity40. Therefore, the effective resistance for the two cases is different,such that the flow induced by the diaphragm motion encounters a fractionof the actual cavity resistance, which yields in the rigid pistonapproximation:

$\begin{matrix}{{R_{gap}^{\prime} = {R_{gap}\frac{a_{f}^{2}}{a^{2}}}},} & (20)\end{matrix}$

where a_(f) is the radius of the optical fiber 30. The Fabry-Perotcavity 40 and the backchamber 65 are fluid volumes that store potentialenergy, hence can impede the diaphragm movement through a spring effect.This effect can be accounted for by the two acoustic compliances, thecavity compliance (C_(cav)), and the backchamber compliance (C_(bc)):

$\begin{matrix}{{C_{cav} = \frac{\pi \; a_{f}^{2}l}{\rho_{0}c^{2}}},{C_{bc} = \frac{\pi \; a_{bc}^{2}L}{\rho_{0}c^{2}}},} & (21)\end{matrix}$

where a_(bc) and L are the radius and length of the backchamber 65,respectively. See, e.g., L. L. Beranek, Acoustics (American Institute ofPhysics, New York, 1986); and M. Rossi, Acoustics and Electroacoustics(Artech House, Inc., 1988). The cavity compliance can be ignored in thecalculations because its reactance is very large in the frequency rangeof interest, due to the small cavity volume. The relatively large volumeof the backchamber 65, on the other hand, includes its acoustic mass:

$\begin{matrix}{M_{bc} = \frac{\rho_{0}L}{3\pi \; a_{bc}^{2}}} & (22)\end{matrix}$

See, e.g., id. In certain embodiments, the reactance of this mass issmall for low frequencies but can dominate the backchamber impedanceabove the Helmholtz frequency of 27 kHz.

Flow Through Annular Channel Around the Fiber

The optical fiber 30 and the diaphragm-size channel 90 through which itpasses defines an annular opening that connects the optical cavity 40 tothe backchamber 65. The resistance and acoustic mass of these annularchannels can be included in the modeling of the sensor 10. Calculationsyield expressions similar to Eqs. (17) and (18):

$\begin{matrix}{{R_{chan} = {\frac{8\; \mu \; }{\pi \; a^{4}}{f_{R}(ɛ)}}},} & (23) \\{{M_{chan} = {\frac{4\; \rho_{0}}{3\pi \; a^{2}}{f_{M}(ɛ)}}},} & (24)\end{matrix}$

Where l is the length of the annular channel. The terms f_(R)(∈) andf_(M)(∈) are functions of ∈=a_(f)/a.

The profile of the axial pressure flow u through an annular channel witha length l, outer diameter a, and inner diameter a_(f), is describedthrough:

$\begin{matrix}{{\frac{\partial v}{\partial r} = {\frac{P\; a}{2\; \mu \; }\left( {{\kappa^{2}/\xi} - \xi} \right)}},} & (25)\end{matrix}$

where ξ=r/a. See, e.g., R. A. Worth, “Accuracy of the parallel-plateanalogy for representation of viscous flow between coaxial cylinders,”J. Appl. Polym. Sci., vol. 24, 319-328 (1979). The plane ξ=κ correspondsto zero shear stress. Integrating Eq. (25), and using the no-slipboundary conditions u=0 for r=a_(f) and r=a, the axial velocity isobtained as:

$\begin{matrix}{{v = {\frac{P\; a^{2}}{4\; \mu \; }\left\lbrack {\left( {1 - \xi^{2}} \right) - {\left( {1 - ɛ^{2}} \right)\ln \; {\xi/\ln}\; ɛ}} \right\rbrack}},} & (26)\end{matrix}$

where ∈=a_(f)/a. Using P=ūR, the acoustic resistance of an annularchannel yields:

$\begin{matrix}{{R_{chan} = {\frac{8\; {\mu }}{\pi \; a^{4}}{f_{R}(ɛ)}}},{{{where}\mspace{14mu} {f_{R}(ɛ)}} = \frac{\ln \; ɛ}{{\left( {1 - ɛ^{4}} \right)\ln \; ɛ} + \left( {1 - ɛ^{2}} \right)^{2}}}} & (27)\end{matrix}$

Similarly, employing U_(k)=½Mū², the acoustic mass of an annular channelyields:

$\begin{matrix}{{M_{chan} = {\frac{4\rho_{0}}{3\pi \; a^{2}}{f_{M}(ɛ)}}},{{{where}\mspace{14mu} {f_{M}(ɛ)}} = \frac{{6\left( {1 - ɛ^{2}} \right)^{3}} + {9\left( {1 - ɛ^{2}} \right)\left( {1 - ɛ^{4}} \right)\ln \; ɛ} + {4\left( {1 - ɛ^{6}} \right)\ln^{2}ɛ}}{{4\left\lbrack {{\left( {1 - ɛ^{4}} \right)\ln \; ɛ} + \left( {1 - ɛ^{2}} \right)^{2}} \right\rbrack}^{2}}}} & (28)\end{matrix}$

In the limit of a circular channel (a_(f)=0), Eqs. (27) and (28) becomeequivalent to Eqs. (17) and (18), respectively, because

${\lim\limits_{ɛ->0}{f_{R}(ɛ)}} = {{\lim\limits_{ɛ->0}{f_{M}(ɛ)}} = 1.}$

While the surface of the optical fiber 30 can be considered perfectlysmooth, as mentioned earlier the silicon sidewalls etched with DRIE canhave a scalloping structure with a mean height of ˜0.25 μm. Such a roughsurface can increase the flow resistance, which can be modeled throughan increase in the viscosity of water or a decrease in the channeldiameter. Based on measurements and calculations in G. M. Mala and D.Li, “Flow characteristics of water in microtubes,” Int. J. Heat FluidFlow, vol. 20, pages 142-148 (1999); and Y. Hu, C. Werner, and D. Li,“Influence of three-dimensional roughness on pressure-driven flowthrough microchannels,” J. Fluids Eng., vol. 125, pages 871-879 (2003),the scalloping roughness (˜0.25 μm) can increase the flow resistance bymore than about 10%. Therefore, the optimum channel size may be adjustedto compensate for this effect.

Modeling Results

Sensor System Response

In the example embodiment, the response of a first sensor (e.g.,300-μm-diameter diaphragm) over the frequency range of 1 Hz-100 kHzcalculated with the lumped-element model is shown in FIG. 35A. Thestructural parameters of the first sensor design are summarized in TableI.

TABLE I Structural dimensions of an example sensor within the sensorsystem Parameter Value Symbol Diaphragm radius (largest) 150 μm a(intermediate) 106 μm (smallest) 75 μm Diaphragm thickness 500 nm h Holeradius 322 nm a_(hole) PC radius 25 μm a_(PC) PC fill factor 0.50 ACavity length 25 μm l Fiber radius 62.5 μm a_(f) Channel length 100 μm lBackchamber radius 3 mm a_(bc) Backchamber length 15 mm L

At low frequencies, with a high-pass cutoff at 25 Hz, water tends toflow through the holes of the photonic-crystal reflective element 22instead of moving the diaphragm 20. The first sensor is insensitive tohydrostatic pressure variations, so that it can be used in, for example,deep-sea applications. At ˜10 kHz, there is a resonance determined bythe diaphragm mechanics and the additional water mass moving with it.The water mass can increase the effective mass of the diaphragm 20 byabout 60 times, so that the resonance drops compared to operation inair. The resonance frequency can be determined from the high-frequencyportion of the acoustic circuit in FIG. 34 as w₀=(M₀C₀)^(−1/2), whereM₀=M_(rad)+M_(dia)+M_(chan)+M_(bc) and 1/C₀=1/C_(dia)+1/C_(bc). Betweenthe cutoff and the resonance there is a wide useful flat band where mostof the incident pressure drops across the diaphragm 20.

As described herein, the shared backchamber 65 allows cross couplingbetween sensors within the sensor system 200. As shown in FIG. 35A, inthe form of an additional resonant feature at about 16 kHz. Thisfrequency corresponds to the resonance of the second sensor (e.g., 212μm diameter), and hence the resonant feature is a result of crosscoupling from this sensor. There is no coupling from the third sensor(e.g., 150 μm diameter), which has a resonance at 23 kHz. The resonancefrequency of the third sensor is substantially close to the Helmholtzfrequency of the backchamber, which isw_(H)=(M_(bc)C_(bc))^(−1/2)=√{square root over (3)}c/L corresponding to27 kHz. As disclosed herein, by designing the backchamber 65 andparallel sensors such that the resonances coincide to the vicinity ofthe Helmholtz resonance, coupling can be suppressed.

One limitation of the lumped modeling is that it does not account forthe acoustic resonances that appear inside the backchamber 65 aboveω=πc/L, corresponding to about 50 kHz. These resonances affect thebackchamber impedance such that it fluctuates from a low value to a highvalue in the vicinity of the resonance frequency. See, e.g., L. L.Beranek, Acoustics (American Institute of Physics, New York, 1986). Thiseffect is not nearly as strong as the reduction of the impedance by theHelmholtz resonance. Although the variation of the backchamber impedancehas a secondary effect on the sensor response, these resonances may bevisible in the actual response spectrum, and hence are not desirable incertain embodiments. Such resonances can be reduced by similar methodsused in loudspeaker enclosures: e.g., by lining the backchamber 65 withsound absorbing or impedance matching layers, so that standing waves aresuppressed. The methods may be modified due to the small size of thebackchamber 65 relative to typical loudspeaker enclosures.

Thermal Noise

FIG. 35B shows the total thermal noise (at 20° C.) transferred to thediaphragm 20, along with contributions from several dissipationchannels, which are radiation loss (dashed line), flow through the holesof the photonic-crystal reflective element 22 (dotted line), and flowthrough the optical cavity 40 and annular channels (dash-dotted line).At low frequencies, the highly dissipative flow through the small holesof the photonic-crystal reflective element 22 can dominate the noisefloor. Above about 1 kHz, the flow through the holes of thephotonic-crystal reflective element 22 can be reduced substantially, sothat the dissipation through the annular channels can dominate thenoise. Above about 40 kHz, the motion of the diaphragm 20 re-radiatesmore energy than lost through other channels, so that the radiation losscan dominate the noise floor. A radiation-loss-limited noise floor isthe fundamental minimum such a sensor system 200 can reach. FIG. 35Cshows the total noise, along with the contributions from the secondsensor (dashed line), and the third sensor (dotted line). The noisecontribution from the second and third sensors is minimum at about 27kHz, because the backchamber 65 is at its Helmholtz resonance, andprevents cross coupling between sensors, as explained above. A typicaloptoelectronic noise spectrum encountered in actual measurements isshown (dash-dotted line) for an optical finesse of ˜10. The noise has awhite-noise component dominated by the relative intensity noise (RIN) ofthe laser (−155 dB/Hz), and by a 1/f-noise component below 1 kHz.

Minimum Detectable Pressure

The noise on the diaphragm 20 normalized to the response yields theminimum detectable pressure (MDP) shown in FIG. 36A. The MDP curve showsthat there is substantially no resonance in the sensitivity in certainembodiments. Resonance effects can be cancelled out, other than thesmall resonant feature due to crosstalk. Because the noise floor is setby the thermal-mechanical noise of the sensor (self noise), the noise atresonance can be amplified too. Through design, the compliance of thesensor system 200 can be adjusted to a high value, so that self noise isdominant over optoelectronic noise. Although increasing the compliancemakes the sensor system 200 more susceptible to Brownian motion, itincreases the signal too. This can make the signal-to-noise ratio (SNR)ultimately larger compared to the case when the noise floor is set bythe optoelectronic noise. The fundamental limit of the SNR can bereached by employing this method. In one way, the sensitivity of thesensor system 200 is increased by making the sensor system 200 noisier.Since in this case the signal and noise are from the same source(acoustic), the resonances in the noise and signal cancel out, so thatno peak in the MDP is observed in FIG. 36A. In this embodiment, the MDPcurve was optimized to match the minimum ambient noise level of theocean by tuning various parameters such as the channel lengths,backchamber volume, and number of holes in the photonic-crystalstructure (see Table I). The match between the calculated MDP curve andthe ocean noise gives this sensor system 200 one of the highest possiblesensitivity over a very wide frequency range of at least 1 Hz-100 kHz.An even better match can be obtained, as shown in FIG. 36B, when onlyone sensor is employed.

Dynamic Range

Among the three diaphragms 20 within the sensor system 200, the largestdiaphragm 20 (e.g., 300 μm diameter) is generally the most fragile one.Therefore, the pressure range of safe operation for the sensor system200 may be limited by the fracture strength of this diaphragm 20. Themaximum pressures the sensor system 200 can be exposed to withoutdamaging the diaphragm 20 is ˜1 MPa (240 dB re. 1 μPa), for a 1 GPayield strength, (see, e.g., W. N. Sharpe, Jr., K. Jackson, K. J. Hemker,and Z. Xie, “Effect of specimen size on Young's modulus and fracturestrength of polysilicon,” J. Micromech. Syst., vol. 10, pages 317-326(2001)), and assuming the holes of the photonic-crystal reflectiveelement 22 do not act as crack-propagation points. However, in certainembodiments, at such large pressures it may be challenging to calibratethe sensor system 200 due to turbulent flow and possible cavitation. Incertain embodiments, cavitation effects may also damage the sensorsystem 200 at lower pressures than the fracture limit of the diaphragm20, reducing the maximum safe pressure. In seawater, cavitation canoccur at pressures as low as about 0.18 MPa (measured at about 10 kHz ata depth of 10 m). See, e.g., V. A. Akulichev and V. I. Il'ichev,“Acoustic cavitation thresholds of sea water in different regions of theworld ocean,” Acoust. Phys., vol. 51, pages 128-138 (2005). The maximumsafe pressure can be reduced to ˜220 dB.

For high-performance applications, the limiting factor in the dynamicrange, in certain embodiments, may be the linearity of the sensor systemresponse. FIG. 37A shows the calculated linearity of the optical signaland the diaphragm displacement. Because the values are normalized, theyare independent of the diaphragm size. S_(FP) is the optical signalamplitude from the Fabry-Perot optical cavity 40. In the linear regime,this amplitude is proportional to the diaphragm displacement amplitudeu₀ through a constant σ_(FP), such that S_(FP)=σ_(FP)u₀. The plot inFIG. 37A assumes an optical finesse of ˜10 (referring to the finesse ofa fiber Fiber-Fabry interferometer, which is different from the finesseof a free-space Fabry-Perot cavity. See, e.g., O. Kilic, M. Digonnet, G.Kino, and O. Solgaard, “Asymmetrical spectral response in fiberFabry-Perot interferometers,” J. Lightwave Technol., vol. 28, pages5648-5656 (2009). Although Fabry-Perot detection provides the highdisplacement sensitivity to detect small pressure amplitudes, itslinearity may be limited. For pressure amplitudes of only ˜5 nm, thelinearity of the Fabry-Perot optical cavity 40 can drop to 90%. Such anonlinearity in certain embodiments can cause harmonic distortion in thesensor system signal. Although the factors for the linearity of thesensor system response can vary depending on the specific application,the sensor system dynamic range for certain embodiments disclosed hereinis calculated for a total harmonic distortion (THD) of about −30 dB. Todetermine the THD for a given pressure, the amplitude of a pure sinewave is distorted with the linearity curves of FIG. 37A. A Fouriertransform of this distorted wave yields the power spectrum of theharmonics. The THD is calculated by dividing the total power in higherharmonics to the power in the fundamental harmonic.

For the first sensor within the sensor system 200, a pressure amplitudeof about 0.6 Pa (115 dB) introduces a THD of about −30 dB as shown inFIG. 37B. The minimum pressure the first sensor can detect in a 1-Hzbandwidth is ˜10 μPa (20 dB). Therefore, the first sensor can addresspressures limited to the range of about 20 dB to 115 dB. As disclosedherein, it is possible to increase this dynamic range by utilizing asecond sensor and a third sensor. Although all three sensors within thesensor system 200 measure the same acoustic signal, they are opticallydecoupled. Therefore, the optical parameters, such as finesse, can bevaried for the second and third sensors without compromising the highsensitivity of the first sensor. The optical finesse of the secondsensor can be reduced to ˜1, corresponding essentially to two-beaminterference. The smaller compliance and reduced finesse allow detectionof larger signals at the expense of sensitivity, providing a pressurerange of about 35 dB to 140 dB for this sensor.

As disclosed herein, the optically decoupled sensors within the sensorsystem 200 allow even greater freedom in tailoring the optical detectionschemes. For example, in certain embodiments, the third sensor does notrequire a high displacement sensitivity, since it is designed to measurelarge signals. Therefore, as described herein, another optical detectionscheme that has less sensitivity but more linearity than the Fabry-Perotdetection can be employed. For example, an optical fiber without areflective element on its end is used, so that there is no significantreflection from its end face (silica-water interface reflection is lessthan 0.3%). In this embodiment, optical interference is prevented. Thediaphragm displacement is detected instead by measuring the opticalpower coupled back into the fiber. This coupling changes with thespacing of the optical cavity 40 because of the diffraction of the lightemerging from the tip of the optical fiber 30. See, e.g., O. Kilic, M.Digonnet, G. Kino, and O. Solgaard, “Asymmetrical spectral response infiber Fabry-Perot interferometers,” J. Lightwave Technol., vol. 28,pages 5648-5656 (2009). In the linear regime, the signal couplingamplitude is proportional to the diaphragm-displacement amplitudethrough a constant σ_(c), such that S_(c)=σ_(C)u₀. With this detectionscheme, the limiting factor can be the linearity of the diaphragmdisplacement, as shown in FIG. 37A. Due to the poor sensitivity of thisscheme, the minimum displacement that the third sensor can measure islimited by the RIN. This is in contrast to the Fabry-Perot detectionemployed in the first sensor and the second sensor, where the limitationis mainly the self noise of the sensor. The third sensor within thesensor system 200 can detect pressures in the range of about 80 dB to180 dB. Therefore, this example demonstrates as disclosed herein, withthe utilization of parallel sensors, the sensor system 200 can becapable of a dynamic range of about 160 dB (20 dB to 180 dB), limited incertain embodiments, only by the linearity of the diaphragm displacementwith pressure.

The dynamic ranges of the first sensor and the third sensor can overlapby about 35 dB (80 dB to 115 dB). Therefore, in certain embodiments, thesecond sensor may not be utilized for applications utilizing a THD of−30 dB. However, for THD levels below −65 dB, the dynamic ranges of thefirst sensor and the third sensor may not overlap at all because theslopes of the THD curves as shown in FIG. 37B for the first sensor andthe third sensor are substantially different. For the power-couplingdetection, the change in THD with respect to the pressure amplitude maybe twice as fast as for the Fabry-Perot-detection (about 30 dB/16 dB vs.about 30 dB/32 dB), so that the overlap between the dynamic ranges cangradually decrease for lower THD. As a result, for applicationsutilizing a THD of around −60 dB or better, the second sensor within thesensor system 200 can be used so that there is sufficient overlapbetween the dynamic ranges. As an example, the dynamic ranges for a −70dB THD are about 20 dB-75 dB, about 35 dB-100 dB, and about 80 dB-160 dBfor the first sensor, the second sensor, and the third sensor,respectively.

Under certain conditions, the lower and upper limits of the pressureranges can be different. For the lower limits, a 1-Hz detectionbandwidth can be assumed. Therefore, for larger bandwidths, the MDP foreach sensor can be increased, hence the dynamic range can be reduced.This reduction also can reduce the overlap in the pressure ranges of theparallel sensors within the sensor system 200. As an example, even for alarge measurement bandwidth of about 100 Hz, there is still an overlapof about 15 dB between the first sensor and the third sensor in a −30 dBTHD regime. However, for a slightly more stringent THD of better than−40 dB, the overlap may not be sufficient so that the second sensor canbe used also to cover the complete dynamic range. For the upper limits,it is assumed that no turbulent flow occurs, so that the analyticalmodel based on laminar flow is still valid. Turbulent flow can occur inmicrofluidic channels for Reynolds numbers (Re) larger than ˜1500. See,e.g., K. V. Sharp and R. J. Adrian, “Transition from laminar toturbulent flow in liquid filled microtubes,” Exp. Fluids, vol. 36, pages741-747 (2004); and C. Rands, B. W. Webb, and D. Maynes,“Characterization of transition to turbulence in microchannels,” Int. J.Heat Mass Transfer, vol. 49, pages 2924-2930 (2006).

An advantage of the analytical model described herein is that it allowsthe calculation of the flow rate through each diaphragm-sized channel90. Since the Reynolds numbers are proportional to the flow rate, it ispossible to analyze various parts of the sensor system 200 to obtain theflow characteristics. The first places for turbulent flow to set on arethe annular channels (e.g., diaphragm-size channels 90), because theycan accommodate all the flow (unlike, e.g., the optical cavity 40)despite their relatively small hydraulic diameters. FIG. 38 shows theReynolds number for the three flow channels 90 for a constant pressureof about 180 dB incident on the third sensor of the sensor system 200.The Reynolds numbers were calculated at different frequencies, and theincident pressure was varied so that the pressure on the smallestdiaphragm (e.g., 150 μm diameter) was constant at the maximum assumedrange of the sensor system 200 (about 180 dB).

In this embodiment, the results shown in FIG. 38 indicate that withinthe dynamic range of the sensor system 200 no turbulent flow isexpected, hence the laminar-flow model and the upper limits of thepressure ranges it predicts are valid. In certain embodiments, thedynamic range cannot be increased substantially because of turbulence.Even with more linear diaphragm structures and displacement sensingmechanisms, the dynamic range can be ultimately limited by turbulentflow.

Experimental Characterization of Example Optical Acoustic Sensor System

The example optical sensor system 200 was characterized inside acontainer filled with distilled water, in the setup shown in FIG. 39.The optical sensor system 200 was interrogated by a fiber-coupled laserwith a wavelength of ˜1550 nm. The laser light first passed through anoptical circulator, which fed the light to the optical sensor system 200and directed the reflected light from the sensor system 200 to aphotoreceiver (e.g., New Focus 2053-FC). The photoreceiver consisted ofan indium-gallium-arsenide PIN photodiode, a gain stage set to 10, and ahigh-pass filter set to 10 Hz.

The optical sensor system 200 was calibrated with a reference sensorsystem (e.g., Celesco LC-10). The reference sensor system had alead-zirconate-titanate reflective element 22, with a calibratedsensitivity of about 39.8 μV/Pa in a wide frequency range of about 0.1Hz to 120 kHz. The reference sensor system 200 was connected to alow-noise preamplifier (e.g., Ithaco 1201) with a gain of about 10 and ahigh-pass cutoff of about 10 Hz.

The electrical outputs of the two sensor systems were connected to adynamic signal analyzer (DSA) (e.g., HP 3562A), which converted the rawsignal into various data such as frequency response, coherence, noisespectrum, and total-harmonic distortion. The DSA also had a built-insignal source that was used to drive the sound source. The drive signalfrom the DSA was fed to a wideband power amplifier (e.g., Krohn-Hite7500) connected to the sound source. The sound source was an acousticprojector consisting of a rigid circular piston (e.g., USRD C100) with adiameter matching the container diameter of 20 cm. Sound was generatedby moving the water column in the cylinder-shaped container up and down.The measured signal from the reference sensor system was fed through aninternal feedback circuit in the DSA to the signal source tocontinuously adjust the output of the sound source. This was done tokeep the pressure amplitude incident on the sensor systems at a constant1 Pa throughout the frequency range. A constant incident pressureprovided a smoother frequency response for both sensor systems, yieldinga more accurate calibration of the optical sensor system 200.

The two sensor systems were mounted on a vibration-isolation stage thatcomprised of a metal plate resting on a slightly deflated air-filledrubber cushion with a torus shape. The metal container was in the formof a plane-wave tube with a height of about 56 cm. The cutoff frequencyof the first cross mode was expected to be ˜4 kHz. Therefore,standing-wave resonances were present in the tube above this frequency.Any effect these resonances could have on the calibration process wassuppressed in two ways: The two sensor systems were mounted close toeach other (<1 cm distance), and for higher frequencies, an additionalmetal tube with a smaller diameter of 2.5 cm was used in the setup. Thetube was covered on the outside with a standard pipe-heat isolationmaterial consisting of 0.95-cm-thick polyethylene with closed airpockets. The isolation material provided a good acoustic isolation fromthe container resonances, due to the large impedance mismatch betweenair pockets and water. The smaller diameter of the tube provided ahigher cross-mode cutoff of ˜35 kHz, yielding a smoother response forfrequencies above about 1 kHz.

The coherence between the reference and optical sensor system spectra,measured with the DSA, is shown in FIG. 40. FIG. 40 indicates that thetwo sensor system signals are strongly correlated from ˜150 Hz to ˜15kHz. The weak correlation above ˜10 kHz suggests that the optical sensorsystem signal is dominated by noise. FIG. 41A shows the measuredfrequency response of the optical sensor system 200. The frequencyresponse is calculated by the DSA by dividing the power spectrum of theoptical sensor system 200 (in units of V), to the power spectrum of thecalibrated reference sensor system (in units of Pa). The response has aresonance at ˜2.2 kHz. Above the resonance frequency, the responsegradually drops, approaching the noise level above ˜10 kHz, so that thecoherence degrades.

The resonance for the sensor system 200 occurs at a rather lowfrequency, deviating from the calculated values (2.2 kHz instead of 10kHz). Among various reasons, such as a slightly larger and less stiffdiaphragm 20 caused by fabrication errors, an important reason asdescribed herein is trapped air in the backchamber 65. The exact size ofthe air bubble was not measured, but visually estimated to be on theorder of about 1-2 mm through a semitransparent part of the sensor head.The theoretical fit in FIG. 41A was obtained with the analytical modelfor an air bubble with an equivalent radius of 1 mm. FIG. 41B shows theexperimental MDP of the sensor system 200 with a theoretical fit alsoobtained with the model. As mentioned above, the sensor system 200 isable to measure pressures as low as 3.5 μPa/Hz^(1/2) for a frequencyrange of 100 Hz to 10 kHz provided by the increased compressibility inthe backchamber 65 caused by the trapped air.

To measure the linearity of the sensor system response, the acousticsource was driven at 200 Hz, and the power spectrum of the opticalsensor system 200 was measured. The incident pressure at 200 Hz wasmeasured as 4 Pa with the calibrated reference sensor system. FIG. 42shows the measured power spectrum of the optical sensor system 200. Itshows that the signal from the fundamental harmonic is substantiallystrong, despite the relatively large incident power (e.g., saturation isweak). The DSA measured a THD of −29 dB, proving that the response ofthe sensor is very linear.

Various embodiments have been described above. Although the inventionhas been described with reference to these specific embodiments, thedescriptions are intended to be illustrative of the invention and arenot intended to be limiting. Various modifications and applications mayoccur to those skilled in the art without departing from the true spiritand scope of the invention as defined in the appended claims.

What is claimed is:
 1. A method of fabricating a sensor, the methodcomprising: providing a movable element; positioning an optical fiberrelative to the movable element to form an optical cavity such thatlight propagates in the optical cavity between the optical fiber and themovable element and is reflected by the movable element, the opticalcavity comprising a medium having a refractive index change withtemperature; and positioning an element within the optical cavity, theelement having a coefficient of thermal expansion and a thickness thatcompensate the refractive index change with temperature.
 2. The methodof claim 1, wherein the movable element comprises a reflective elementand a diaphragm, the medium comprises water, and the element within theoptical cavity comprises silica and has a thickness approximately equalto a distance between the optical fiber and the movable element.
 3. Themethod of claim 2, wherein the element within the optical cavitycomprises the diaphragm.
 4. The method of claim 1, wherein the movableelement comprises a reflective element and a diaphragm, the mediumcomprises water in a region between the optical fiber and the movableelement, and the element within the optical cavity comprises silica andhas a thickness approximately equal to a thickness of the region betweenthe optical fiber and the movable element.
 5. The method of claim 1,wherein the element within the optical cavity comprises a portion of theoptical fiber.
 6. The method of claim 1, wherein the element within theoptical cavity is mechanically coupled to the optical fiber.
 7. Themethod of claim 1, wherein the optical fiber comprise a secondreflective element, the second reflective element and the reflectiveelement forming a Fabry-Perot cavity therebetween.
 8. The method ofclaim 7, wherein the element within the optical cavity is between thesecond reflective element and the reflective element.
 9. The method ofclaim 1 wherein the sensor has a reduced sensitivity to temperaturevariations as compared to a sensor without the element within theoptical cavity.
 10. A method of fabricating an acoustic sensor, themethod comprising: providing a diaphragm comprising a reflectiveelement; positioning an optical fiber relative to the reflective elementsuch that light emits from the optical fiber and is reflected from thereflective element, wherein positioning the optical fiber relative tothe reflective element comprises forming an optical cavity therebetween;and mechanically coupling the diaphragm to the optical fiber with astructural element, wherein the structural element comprises silica. 11.The method of claim 10, wherein the reflective element comprises aphotonic-crystal structure.
 12. The method of claim 10, wherein thediaphragm comprises silica.
 13. The method of claim 12, wherein thediaphragm has a thickness approximately equal to a distance between theoptical fiber and the diaphragm.
 14. The method of claim 10, furthercomprising positioning an element comprising silica within the opticalcavity.
 15. The method of claim 14, further comprising selecting athickness for the element comprising silica approximately equal to adistance between an end of the optical fiber and the diaphragm.
 16. Themethod of claim 10, further comprising silicate bonding the diaphragm tothe structural element.
 17. The method of claim 10, further comprisingpositioning a compensating element comprising silica within the opticalcavity and spaced from the diaphragm.
 18. The method of claim 10,wherein an end of the optical fiber comprises a second reflectiveelement, the second reflective element and the reflective elementforming a Fabry-Perot cavity therebetween.
 19. The method of claim 8,wherein the optical cavity comprises a liquid, and the method furthercomprising positioning at least one gas bubble within a volume of theliquid to increase sensitivity.